############################################################################### # # # http://www.research.att.com/~njas/sequences/gammoids.txt # # # # gammoids.txt -- Maple source file for handling gammiers and gammoids, # # mathematical structures invented by Pierre Lamothe # # for the containment of musical scales. # # See his "Musique et Science" site # # at http://www.aei.ca/~plamothe/ # # Used to compute OEIS sequences A090000 - A090072. # # (to be renumbered...) # # # # This module coded by Antti Karttunen May 2001 & Jun 2003, # # and released into Public Domain. # # # # The code has been tested in Maple V Release 4, Version 4.00c, Jun 7 1996. # # # # Last edited 14. August 2003 # # # ############################################################################### with(linalg); # For matrix operations. with(numtheory); # For divisors, ifactors etc. ############################################################################### # # # Useful (?) functions from findnext.txt # # # ############################################################################### # A000120(0) = 0, A000120(2n) = A000120(n), A000120(2n+1) = A000120(n) + 1. A000120 := proc(nn) local n,s; n := nn; s := 0; while(n > 0) do s := s + (n mod 2); n := (n-(n mod 2))/2; od; RETURN(s); end; A000523 := proc(n) local nn,i; # Was: floor_log_2 nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end; A062383 := proc(n) option remember; if(0 = n) then RETURN(1); else RETURN(2*A062383(floor(n/2))); fi; end; A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1); # The X-projection & A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2); # Y-projection (of A001477) absdiff := (x,y) -> abs(x-y); divby := (n,d) -> n/d; avg := a -> (convert(a,`+`)/nops(a)); nthmember := proc(e,l) local k; if(member(e,l,'k')) then RETURN(k) else RETURN(0); fi; end; positions := proc(e,ll) local a,k,l,m; l := ll; m := 1; a := []; while(member(e,l[m..nops(l)],'k')) do a := [op(a),(k+m-1)]; m := k+m; od; RETURN(a); end; negative_terms := proc(b) local a,i; a := []; for i from 1 to nops(b) do if(b[i] < 0) then a := [op(a),i]; fi; od; RETURN(a); end; palindrome_lists := proc(b) local a,i; a := []; for i from 1 to nops(b) do if(reverse(b[i]) = b[i]) then a := [op(a),i]; fi; od; RETURN(a); end; car:=proc(l) if 0 = nops(l) then ([]) else (op(1,l)): fi: end: cdr:=proc(l) if 0 = nops(l) then ([]) else (l[2..nops(l)]): fi: end: last_term:=proc(l) local n: n := nops(l); if(0 = n) then ([]) else (op(n,l)): fi: end: cons:=proc(head,tail) ([head, op(tail)]); end: list3:=proc(el1,el2,el3) ([el1,el2,el3]); end: ############################################################################## ############################################################################## # # # The main code for computing OEIS sequences A090000 - A090035 # # # ############################################################################## # There are better ways to do this, see: # http://www.research.att.com/projects/OEIS?Anum=A062383 coerce1st_octave := proc(r) option remember; if(r < 1) then coerce1st_octave(2*r); else if(r >= 2) then coerce1st_octave(r/2); else (r); fi; fi; end; coerce1st_octave_v2 := n -> `if`(((not type(n,integer)) or (not 1 = (n mod 2))),`use only for odd numbers!`,n/A062383((n-1)/2)); # `if`(([seq(coerce1st_octave(2*n+1),n=0..128)]=[seq(coerce1st_octave_v2(2*n+1),n=0..128)]),`samat`,`erit`); --> samat PartProducts1 := proc(a) local b,e,p; p := 1; b := [1]; for e in a do p := p*e; b := [op(b),p]; od; RETURN(b); end; FirstRatios := proc(a) local b,e,i; b := []; for i from 1 to nops(a)-1 do b := [op(b),(a[i+1]/a[i])]; od; RETURN(b); end; # Converts IB-code to "harmonic generator", # i.e. each 1 in bit-position n stands for # the odd number 2n+1: # IB2HG(151); --> [1, 3, 5, 9, 15] # IB2HG(183); --> [1, 3, 5, 9, 11, 15] IB2HG := proc(ib) local a,x,i; a := []; x := ib; i := 1; while(x > 0) do if(1 = (x mod 2)) then a := [op(a),i]; fi; x := floor(x/2); i := i+2; od; RETURN(a); end; # Compute all ratios (intervals) obtained with # cartesian product from the harmonic generator, # discard the duplicates and sort: # HG2Intervals(IB2HG(151)); # --> [1, 16/15, 10/9, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, 9/5, 15/8] HG2Intervals := proc(hg) local a,i,j,n; a := []; n := nops(hg); for i from 1 to n do for j from i to n do a := [op(a),coerce1st_octave(hg[j]/hg[i]),coerce1st_octave(hg[i]/hg[j])]; od; od; RETURN(sort(convert(convert(a,set),list))); end; # Note: it shouldn't matter whether we coerce the ratios to [1,2[ before or after # discarding the duplicate ones, if we are only concerned about their number # as in A090042: HG2Ratios := proc(hg) local a,i,j,n; a := []; n := nops(hg); for i from 1 to n do for j from i to n do a := [op(a),(hg[j]/hg[i]),(hg[i]/hg[j])]; od; od; RETURN(sort(convert(convert(a,set),list))); end; # See http://groups.yahoo.com/group/tuning/message/21703 # (and http://groups.yahoo.com/group/tuning/message/20746) # Let I be an ordered set of intervals in the first octave. By definition, an # "atom" of I is any interval k greater than unison that cannot be splitted # in two other intervals x and y greater than unison, where x < k and y < k. # Would the concept of atom survive in the quotient group (Q+,*)/<2^-1,2^+1> # without recurring to harmoid's/gammoid's pre-image in (Q+,*) ??? # (How to interpret "less" or "greater than" in that quotient group? # Cf. how to interpret "divisibility" in Zn rings? No, doesn't seem to.) # AllAtoms(HG2Intervals(IB2HG(151))); -> [16/15, 10/9, 9/8] AllAtoms := proc(is) local s,a,i,j,n; s := cdr(is); n := nops(s); a := [s[1]]; for i from 2 to n do for j from 1 to (i-1) do if(member((s[i]/s[j]),s)) then break; fi; od; if(j > (i-1)) then a := [op(a),s[i]]; fi; od; RETURN(a); end; order_in_octave := (a,b) -> `if`((coerce1st_octave(a) < coerce1st_octave(b)),true,false); sort_by_order_in_octave := a -> sort(a,order_in_octave); # # HG2Matrix(IB2HG(151)); # --> # # [ 1 9/8 5/4 3/2 15/8] # [ ] # [16/9 1 10/9 4/3 5/3 ] # [ ] # [8/5 9/5 1 6/5 3/2 ] # [ ] # [4/3 3/2 5/3 1 5/4 ] # [ ] # [ ] # [ 16/15 6/5 4/3 8/5 1 ] # [ ] # HG2Matrix := proc(h) local hg,a,b,i,j,n; a := []; hg := sort_by_order_in_octave(h); n := nops(hg); for i from 1 to n do b := []; for j from 1 to n do b := [op(b),coerce1st_octave(hg[j]/hg[i])]; od; a := [op(a),b]; od; RETURN(array(1..n,1..n,a)); end; # HG2AM creates an Adjacency Matrix for this harmonic generator. # Here there's 1 in pos AM(row,col) if one can hop from col:th interval in # HG2Intervals(IB2HG(151)): # [1, 16/15, 10/9, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, 9/5, 15/8] # to the row:th interval in the same list, using one of the # AllAtoms(HG2Intervals(IB2HG(151))); -> [16/15, 10/9, 9/8] # as a "hopping step", either in the rows of HG2Matrix(IB2HG(151)); (shown above) # when going from left to right, or in columns, when going from # bottom to top. # # AM := HG2AM(IB2HG(151)); # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,1 ,1 ,1] # [ ] # [1 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [1 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [1 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,1 ,0 ,1 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,0 ,1 ,1 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # AM := [0 ,0 ,0 ,0 ,1 ,1 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,1 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,1 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,1 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,1 ,1 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,1 ,0 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,1 ,0 ,0 ,0] # # R := multiply(AM,multiply(AM,multiply(AM,multiply(AM,multiply(AM,multiply(AM,AM)))))); # or just: R := evalm(AM ^ 7); # # R, which is the degreeth power of an adjacency matrix AM, # then gives the number of possible paths between the intervals of # HG2Intervals(IB2HG(151)) # [1, 16/15, 10/9, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, 9/5, 15/8] # in the corresponding gammier. The top-left corner R[1,1] tells # that we have altogether 16 ways to traverse from the unison # to the octave, i.e. 16 different modes. See the function ComputeModes below. # # [16,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,4 ,4 ,8 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,4 ,4 ,8 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,4 ,4 ,8 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,8 ,8 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,8 ,8 ,0 ,0 ,0 ,0 ,0 ,0 ,0] # [ ] # R := [0 ,0 ,0 ,0 ,0 ,0 ,16,0 ,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,16,0 ,0 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,8 ,8 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,8 ,8 ,0 ,0 ,0] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,8 ,8 ,8] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,4 ,4 ,4] # [ ] # [0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,4 ,4 ,4] # HG2AM := hg -> HG2AM_aux1(hg,HG2Intervals(hg)); HG2AM_aux1 := (hg,is) -> HG2AM_aux2(hg,is,AllAtoms(is)); HG2AM_aux2 := proc(hg,is,as) local am,hm,n,c,r,x,y,xpos,ypos; n := nops(is); hm := HG2Matrix(hg); am := array(1..n,1..n); for r from 1 to n do for c from 1 to n do am[c,r] := 0; od; od; for r from 1 to rowdim(hm) do ypos := nthmember(hm[r,coldim(hm)],is); for c from 1 to coldim(hm) do if(1 = c) then x := hm[r,coldim(hm)]; else x := hm[r,c-1]; fi; y := hm[r,c]; xpos := ypos; ypos := nthmember(y,is); if(1 = y) then y := 2; fi; if(member((y/x),as)) then am[xpos,ypos] := 1; fi; od; od; for c from 1 to coldim(hm) do ypos := nthmember(hm[1,c],is); for r from rowdim(hm) by -1 to 1 # From bottom to top. do if(rowdim(hm) = r) then x := hm[1,c]; else x := hm[r+1,c]; fi; y := hm[r,c]; xpos := ypos; ypos := nthmember(y,is); if(1 = y) then y := 2; fi; if(member((y/x),as)) then am[xpos,ypos] := 1; fi; od; od; RETURN(transpose(transpose(am))); # Avoid bugs in Maple's matrix package... end; # The following function is used to check whether the resulting adjacency # matrix contains any row or column with ONLY zero entries. # And as 1 and 2 are thought as the same node, this means that there # are no isolated vertices or subgraphs, and no "dead starts nor dead ends", # i.e. from every node there is an edge from some other node, and from every # node there's at least one edge leaving to some other node. # (Remember that this is a directed graph) # AND THIS SUFFICES FOR CHECKING # # A) THE CONTIGUITY # B) THAT THERE ARE ANY MODES AT ALL (paths from 1 to 1, we don't even need # to raise the matrix to the degreeth power and check element [1,1] to see # that) # C) THAT ALL INTERVALS BELONG TO A SOME MODE. # # This returns 0 if there were no zero-rows nor zero-columns, # otherwise the first zero row's index, # or the first zero column's index negated: AM_contains_zero_rows := proc(am) local i,r; for r from 1 to rowdim(am) do if(0 = add(i,i=convert(row(am,r),list))) then RETURN(r); fi; od; RETURN(0); end; AM_contains_zero_columns := proc(am) local i,c; for c from 1 to coldim(am) do if(0 = add(i,i=convert(col(am,c),list))) then RETURN(c); fi; od; RETURN(0); end; AM_contains_zero_columns_or_rows := proc(am) local s; s := AM_contains_zero_rows(am); if(0 <> s) then RETURN(s); fi; s := AM_contains_zero_columns(am); if(0 <> s) then RETURN(-s); fi; RETURN(0); end; list_as_binary := proc(a) local s,b; s := 0; for b in a do s := 2*s + b; od; RETURN(s); end; # As above, but prepended with 1-bit: (so A000523 of the result is a square number) list_as_binary1 := proc(a) local s,b; s := 1; for b in a do s := 2*s + b; od; RETURN(s); end; binexp2list := proc(nn) local a,n; n := nn; a := []; while(n > 0) do a := [(n mod 2),op(a)]; n := (n-a[1])/2; od; RETURN(a); end; squarebinexp2lists := proc(nn) local a,b,n,i,r; n := nn; r := sqrt(A000523(n)); if((not type(r,integer)) or (0 = r)) then RETURN([]); fi; i := 0; a := []; while(n > 0) do if(0 = (i mod r)) then if(i > 0) then a := [b,op(a)]; fi; b := []; fi; b := [(n mod 2),op(b)]; n := (n-b[1])/2; i := i+1; od; RETURN(a); end; AM2list := am -> map(op,map(convert,[row(am,1..rowdim(am))],list)); # AMcertificate(HG2AM(IB2HG(151))); # --> 685084468365777036177292784393058930488500158472 # This one for internal use: AMcertificate := am -> list_as_binary(AM2list(am)); # This one for sequences: AMcertificate1 := am -> list_as_binary1(AM2list(am)); FirstNonRegularAtom := proc(as) local a,b; b := []; for a in as do if(a >= (2/a)) then RETURN(a); fi; od; RETURN(0); end; # This gives the shrutal basis from tonal basis: # (Should we make sure that FirstRatios(tb) do not # contain duplicates?) # TB2SB(AllAtoms(HG2Intervals([1,3,5,9,15]))); -> [16/15, 25/24, 81/80]; TB2SB := tb -> [tb[1],op(FirstRatios(tb))]; # This gives the primal basis from shrutal or tonal basis, # i.e. the sorted sequence of all primes occurring in the factorizations # of all their ratios. # SBorTB2PB(AllAtoms(HG2Intervals([1,3,5,9,15]))); -> [2, 3, 5] # SBorTB2PB(TB2SB(AllAtoms(HG2Intervals([1,3,5,9,15])))); -> [2, 3, 5] SBorTB2PB := proc(b) local f,a,r,o; a := []; for r in b do f := ifactors(r)[2]; for o in f do a := [op(a),o[1]]; od; od; RETURN(sort(convert(convert(a,set),list))); end; # Actually, just like getprop in Lisp: find_ps_exponent := proc(p,f) local i; for i in f do if(i[1] = p) then RETURN(i[2]); fi; od; RETURN(0); end; # # SB_orTB2Pmatrix(AllAtoms(HG2Intervals(IB2HG(151)))); # --> # [ 4 1 -3] # [ ] # [-1 -2 2] # [ ] # [-1 1 0] # # Atoms in Zarlino's Gammier (151) are 16/15, 10/9 and 9/8 # and thus the column vectors in above matrix give exponents # in their factorizations: # 2^4 * 3^-1 * 5^-1 # 2^1 * 3^-2 * 5^1 # 2^-3 * 3^2 * 5^0 # SB_orTB2Pmatrix := proc(b) local p,a,pb,c,f,r; a := []; pb := SBorTB2PB(b); for r in b do f := ifactors(r)[2]; c := []; for p in pb do c := [op(c),find_ps_exponent(p,f)]; od; a := [op(a),c]; od; # # of rows, # of cols RETURN(transpose(array(1..nops(b),1..nops(pb),a))); end; # ComputeDegree computes the degree of harmoid/gammoid/gammier generated # by the harmonic generator given as its argument. # # Note that this can handle also cases where the tonal matrix # has linearly dependent column vectors, as long as the # variable(s)of the parametric solution given by linsolve # will all be eliminated when its elements are added up. # # E.g., for Arabic+Zarlino gammier 183, we get the following # alternative parametric solutions from linsolve: # # [-1+t1, 3-t1, 3-t1, 2, t1] or [t1, -t1+2, -t1+2, 2, t1+1] # but summing them up leaves only the constant 7. # # Similarly for Gammoid5381, although its primal base to # tonal (and shrutal base) matrices are singular, it still has # a constant degree 5. ( = sum [t1, -t1+2, -t1+2, t1, 1] ) # # However, if after such addition there still remains # a parameter (the sum is not strictly numeric), then # we conclude that the given harmoid is not congruent, # and we return 0 as a result. ComputeDegree := hg -> ComputeDegreeAux(AllAtoms(HG2Intervals(hg))); ComputeDegreeAux := proc(as) local m,v,i,r,s; m := SB_orTB2Pmatrix(as); r := rowdim(m); v := vector(r); v[1] := 1; for i from 2 to r do v[i] := 0; od; s := convert(convert(linsolve(m,v,'r'),list),`+`); if(type(s,numeric)) then RETURN(s); else RETURN(0); fi; end; descending := (a,b) -> `if`(((a) > (b)),true,false); ComputeMinimalExampleForDivisorLattice := ib -> ComputeMEDLaux(AllAtoms(HG2Intervals(IB2HG(ib)))); ComputeMEDLaux := proc(as) local m,v,i,r,s,n; m := SB_orTB2Pmatrix(as); r := rowdim(m); v := vector(r); v[1] := 1; for i from 2 to r do v[i] := 0; od; s := convert(linsolve(m,v,'r'),list); # This check is equal to that the rank and coldim of m (= nops(as)) are same: # for i from 1 to nops(s) # do # if(not type(s[i],numeric)) then RETURN(0); fi; # od; if(rank(m) <> coldim(m)) then RETURN(0); fi; s := sort(s,descending); RETURN(mul((ithprime(i)^s[i]),i=1..nops(s))); end; ComputeModes := proc(am,d) local am_d; am_d := evalm(am ^ d); RETURN(am_d[1,1]); # How many modes we have? end; # For checking duplicates: GLOB_CERTS := VECTOR([]); # Note that we consider two gammoids/gammiers generated by different # harmonic generators equal only if # # a) they generate the same set of intervals (thus, also the same set of atoms) # # and # # b) they have exactly same adjacency matrices, as produced by # the function HG2AM. (we insert its "certificate" into the # front of the interval list before storing it to the GLOB_CERTS # vector here). # check_duplicates := proc(ib,certlist) local j; global GLOB_CERTS; for j from 1 to ib-1 do if(certlist = GLOB_CERTS[j]) then RETURN(j); fi; od; GLOB_CERTS[ib] := certlist; RETURN(0); end; IsGammoidOrGammier := proc(ib,must_be_fertile,dups_filtered) local hg,as,is,d,dup,am,am_d,amhash; if(type(ib,numeric)) then hg := IB2HG(ib); else hg := ib; fi; if(nops(hg) <= 1) then RETURN(`unison harmonic generator`); fi; is := HG2Intervals(hg); # Returns always a sorted list. as := AllAtoms(is); if(0 <> FirstNonRegularAtom(as)) then RETURN(`not regular`); fi; d := ComputeDegreeAux(as); if(0 = d) then RETURN(`not congruous`); fi; if((must_be_fertile > 0) and (d <= nops(hg))) then RETURN(`not fertile`); fi; am := HG2AM_aux2(hg,is,as); amhash := AMcertificate(am); if(0 <> AM_contains_zero_columns_or_rows(am)) then RETURN(`not contiguous`); fi; if((dups_filtered > 0) and type(ib,numeric)) then dup := check_duplicates(ib,[amhash,op(is)]); if(dup > 0) then RETURN(sprintf(`duplicate of %a`,dup)); fi; fi; # RETURN([nops(hg),nops(is),nops(as),d,ComputeModes(am,d)]); RETURN([nops(hg),nops(is),nops(as),d]); end; GLOB_COL := []; # For bulk use. Compute FindGammoids_or_Gammiers(16384,1,1), if possible. FindGammoids_or_Gammiers := proc(upto_n,fertile,dups_filtered) local a,i,c; global GLOB_COL; GLOB_COL := []; for i from 1 to upto_n do c := IsGammoidOrGammier(i,fertile,dups_filtered); if(not type(c,string)) then GLOB_COL := [op(GLOB_COL),i]; print(sprintf(`ib %a: %a`,i,c)); else if(not (`not fertile` = c)) then print(sprintf(`ib %a`,i),c); fi; fi; od; RETURN(GLOB_COL); end; # One-based: nthGammoid_or_Gammier := proc(n,fertile,dups_filtered) option remember; local i,c; if(1 = n) then i := n; else i := nthGammoid_or_Gammier(n-1,fertile,dups_filtered)+1; fi; while(i > 0) # i.e. indefinitely, until the next one is found. do c := IsGammoidOrGammier(i,fertile,dups_filtered); if(not type(c,string)) then RETURN(i); else if(not (`not fertile` = c)) then print(sprintf(`ib %a`,i),c); fi; fi; i := i+1; od; end; A090000 := n -> nthGammoid_or_Gammier(n,0,1); # Gammoids, Dups filtered. A090001 := n -> nthGammoid_or_Gammier(n,0,0); # Gammoids, Dups not filtered. A090002 := n -> nops(IB2HG(A090001(n))); # Number of 1-bits in HG. A090003 := n -> nops(HG2Intervals(IB2HG(A090001(n)))); # Number of Intervals generated. A090004 := n -> nops(AllAtoms(HG2Intervals(IB2HG(A090001(n))))); # Number of Atoms. A090005 := n -> ComputeDegree(IB2HG(A090001(n))); # Degree (pentatonic, heptatonic?) A090006 := n -> ComputeModes(HG2AM(IB2HG(A090001(n))),A090005(n)); A090007 := n -> rank(SB_orTB2Pmatrix(AllAtoms(HG2Intervals(IB2HG(A090001(n)))))); A090002v2 := n -> A000120(A090001(n)); A090003v2 := n -> A090042(A090001(n)); A090004v2 := n -> A090043(A090001(n)); A090005v2 := n -> A090054(A090001(n)); A090007v2 := n -> A090044(A090001(n)); # Pos of nth gammoid where the rank of tonal matrix differs from its coldim (i.e. A090004(n)). # Complement of A090048. A090008 := proc(n) option remember; local i,g,tm; if(1 = n) then i := n; else i := A090008(n-1)+1; fi; while(i > 0) # i.e. indefinitely, until the next one is found. do g := A090001(i); tm := SB_orTB2Pmatrix(AllAtoms(HG2Intervals(IB2HG(g)))); if(not (rank(tm) = coldim(tm))) then RETURN(i); fi; i := i+1; od; end; A090009 := n -> A090001(A090008(n)); # Nth gammoid where the rank differs from coldim. # Just for checking... In range 1..225 all are zero. GammoidAMdeterminants := n -> det(HG2AM(IB2HG(A090001(n)))); # Pos of nth gammier among the gammoids: A090010 := proc(n) option remember; local i,g; if(1 = n) then i := n; else i := A090010(n-1)+1; fi; g := A090011(n); while(i > 0) # i.e. indefinitely, until the next one is found. do if(A090001(i) = g) then RETURN(i); fi; i := i+1; od; end; # Do not submit gammiers with duplicates filtered, because ... A090011 := n -> nthGammoid_or_Gammier(n,1,0); # Gammiers, Dups not filtered. # Where is the first duplicate gammier? # At: A090011(929) = 8338, duplicate of A090011(1)=23 # and: A090011(1023) = 9234, duplicate of A090011(2)=27 A090012 := n -> nops(IB2HG(A090011(n))); # Number of 1-bits in HG. A090013 := n -> nops(HG2Intervals(IB2HG(A090011(n)))); # Number of Intervals generated. A090014 := n -> nops(AllAtoms(HG2Intervals(IB2HG(A090011(n))))); # Number of Atoms. A090015 := n -> ComputeDegree(IB2HG(A090011(n))); # Degree (pentatonic, heptatonic?) A090016 := n -> ComputeModes(HG2AM(IB2HG(A090011(n))),A090015(n)); A090012v2 := n -> A000120(A090011(n)); A090013v2 := n -> A090042(A090011(n)); A090014v2 := n -> A090043(A090011(n)); A090015v2 := n -> A090054(A090011(n)); A090017v2 := n -> A090044(A090011(n)); A090017 := n -> rank(SB_orTB2Pmatrix(AllAtoms(HG2Intervals(IB2HG(A090011(n)))))); # Pos of nth gammier where the rank of tonal matrix differs from its coldim (i.e. A090014(n)). # Complement of A090051. A090018 := proc(n) option remember; local i,g,tm; if(1 = n) then i := n; else i := A090018(n-1)+1; fi; while(i > 0) # i.e. indefinitely, until the next one is found. do g := A090011(i); tm := SB_orTB2Pmatrix(AllAtoms(HG2Intervals(IB2HG(g)))); if(not (rank(tm) = coldim(tm))) then RETURN(i); fi; i := i+1; od; end; A090019 := n -> A090011(A090018(n)); # Nth gammier where the rank differs from coldim. # Pos of the nth gammier with degree d. pos_of_nthgammier_of_degree := proc(n,d) option remember; local i,g,tm; if(1 = n) then i := n; else i := pos_of_nthgammier_of_degree(n-1,d)+1; fi; while(i > 0) # i.e. indefinitely, until the next one is found. do if(A090015(i) = d) then RETURN(i); fi; i := i+1; od; end; nthgammier_of_degree := (n,d) -> A090011(pos_of_nthgammier_of_degree(n,d)); # Offset 1: A090020 := n -> pos_of_nthgammier_of_degree(A025581(n-1)+1,A002262(n-1)+5); A090021 := n -> A090011(A090020(n)); A090022 := n -> pos_of_nthgammier_of_degree(n,5); # Pentatonic, positions. A090023 := n -> A090011(A090022(n)); # Pentatonic gammiers, IB-codes. A090024 := n -> pos_of_nthgammier_of_degree(n,6); # Hexatonic, positions. A090025 := n -> A090011(A090024(n)); # Hexatonic gammiers, IB-codes. A090026 := n -> pos_of_nthgammier_of_degree(n,7); # Heptatonic, positions. A090027 := n -> A090011(A090026(n)); # Heptatonic gammiers, IB-codes. A090028 := n -> pos_of_nthgammier_of_degree(n,8); # Octatonic, positions. A090029 := n -> A090011(A090028(n)); # Octatonic gammiers, IB-codes. A090030 := n -> pos_of_nthgammier_of_degree(n,9); # Nonatonic, positions. A090031 := n -> A090011(A090030(n)); # Nonatonic gammiers, IB-codes. A090032 := n -> pos_of_nthgammier_of_degree(n,10); # Decatonic, positions. A090033 := n -> A090011(A090032(n)); # Decatonic gammiers, IB-codes. A090034 := n -> pos_of_nthgammier_of_degree(n,11); # Enneatonic, positions. A090035 := n -> A090011(A090034(n)); # Enneatonic gammiers, IB-codes. A090036 := n -> pos_of_nthgammier_of_degree(n,12); # Dodecatonic, positions. A090037 := n -> A090011(A090036(n)); # Dodecatonic gammiers, IB-codes. A090042 := n -> nops(HG2Intervals(IB2HG(n))); # Interesting sequence per se. A090042v2 := n -> nops(HG2Ratios(IB2HG(n))); # Easier way. A090043 := n -> `if`((A000120(n) < 2),0,nops(AllAtoms(HG2Intervals(IB2HG(n))))); A090044 := n -> `if`((A000120(n) < 2),0,rank(SB_orTB2Pmatrix(AllAtoms(HG2Intervals(IB2HG(n)))))); A090045 := n -> A090043(n)-A090044(n); # Positions of zeros in A090045, complement of A090047: A090046 := proc(n) option remember; local i; if(0 = n) then i := n; else i := A090046(n-1)+1; fi; while(i >= 0) # i.e. indefinitely, until the next one is found. do if(A090043(i) = A090044(i)) then RETURN(i); fi; i := i+1; od; end; # Positions of non-zeros in A090045, complement of A090046: A090047 := proc(n) option remember; local i; if(0 = n) then i := n; else i := A090047(n-1)+1; fi; while(i >= 0) # i.e. indefinitely, until the next one is found. do if(not (A090043(i) = A090044(i))) then RETURN(i); fi; i := i+1; od; end; # Pos of nth gammoid where the rank of tonal matrix is equal to its coldim # (i.e. A090004(n) = A090007(n)). Complement of A090008. A090048 := proc(n) option remember; local i,g,tm; if(1 = n) then i := n; else i := A090048(n-1)+1; fi; while(i > 0) # i.e. indefinitely, until the next one is found. do g := A090001(i); tm := SB_orTB2Pmatrix(AllAtoms(HG2Intervals(IB2HG(g)))); if(rank(tm) = coldim(tm)) then RETURN(i); fi; i := i+1; od; end; A090049 := n -> A090001(A090048(n)); # Nth gammoid where the rank equals coldim. A090050 := n -> ComputeMinimalExampleForDivisorLattice(A090049(n)); # Pos of nth gammier where the rank of tonal matrix is equal to its coldim # (i.e. A090014(n) = A090017(n)). Complement of A090018. A090051 := proc(n) option remember; local i,g,tm; if(1 = n) then i := n; else i := A090051(n-1)+1; fi; while(i > 0) # i.e. indefinitely, until the next one is found. do g := A090011(i); tm := SB_orTB2Pmatrix(AllAtoms(HG2Intervals(IB2HG(g)))); if(rank(tm) = coldim(tm)) then RETURN(i); fi; i := i+1; od; end; A090052 := n -> A090011(A090051(n)); # Nth gammier where the rank equals coldim. A090053 := n -> ComputeMinimalExampleForDivisorLattice(A090052(n)); A090054 := n -> `if`((A000120(n) < 2),0,ComputeDegree(IB2HG(n))); # Positions of zeros in A090054, complement of A090056: A090055 := proc(n) option remember; local i; if(0 = n) then i := n; else i := A090055(n-1)+1; fi; while(i >= 0) # i.e. indefinitely, until the next one is found. do if(0 = A090054(i)) then RETURN(i); fi; i := i+1; od; end; # Positions of non-zeros in A090054, complement of A090055: # This defines the congruity, A090001 and A090011 are subsets of this sequence. A090056 := proc(n) option remember; local i; if(0 = n) then i := n; else i := A090056(n-1)+1; fi; while(i >= 0) # i.e. indefinitely, until the next one is found. do if(0 <> A090054(i)) then RETURN(i); fi; i := i+1; od; end; # Degree minus # of elements in harmonic generator: A090057 := n -> A090054(n)-A000120(n); # Positions of positive terms in A090057: # This defines the fertility, i.e. A090011 is an intersect of A090001 and A090058. A090058 := proc(n) option remember; local i; if(0 = n) then i := n; else i := A090058(n-1)+1; fi; while(i >= 0) # i.e. indefinitely, until the next one is found. do if(A090057(i) > 0) then RETURN(i); fi; i := i+1; od; end; # No, do not submit this one: characteristic function for the regularity. charfun_for_A090059 := n -> `if`((A000120(n) < 2),0,`if`((0 = FirstNonRegularAtom(AllAtoms(HG2Intervals(IB2HG(n))))),1,0)); # Positions of non-zero terms in charfun_for_A090059: # This defines the regularity, i.e. A090001 and A090011 # are subsets of this sequence: A090059 := proc(n) option remember; local i; if(0 = n) then i := n; else i := A090059(n-1)+1; fi; while(i >= 0) # i.e. indefinitely, until the next one is found. do if(0 <> charfun_for_A090059(i)) then RETURN(i); fi; i := i+1; od; end; # Just for checking, that we have an identity function here: A001477v2 := proc(n) local r; r := sqrt(A000523(n)); if((not type(r,integer)) or (0 = r)) then RETURN(n); fi; RETURN(AMcertificate1(array(1..r,1..r,squarebinexp2lists(n)))); end; # An involution of natural numbers: A090060 := proc(n) local r; r := sqrt(A000523(n)); if((not type(r,integer)) or (0 = r)) then RETURN(n); fi; RETURN(AMcertificate1(transpose(array(1..r,1..r,squarebinexp2lists(n))))); end; # Just for checking, that A090060 is indeed an involution: A001477v3 := n -> A090060(A090060(n)); # Characteristic function for the next one: A090061 := proc(n) local r,am; r := sqrt(A000523(n)); if((not type(r,integer))) then RETURN(0); fi; am := array(1..r,1..r,squarebinexp2lists(n)); if(0 <> AM_contains_zero_rows(am)) then RETURN(0); else RETURN(1); fi; end; A090062 := proc(n) option remember; local i; if(0 = n) then i := n; else i := A090062(n-1)+1; fi; while(i >= 0) # i.e. indefinitely, until the next one is found. do if(0 <> A090061(i)) then RETURN(i); fi; i := i+1; od; end; # Characteristic function for the next one: A090063 := proc(n) local r,am; r := sqrt(A000523(n)); if((not type(r,integer))) then RETURN(0); fi; am := array(1..r,1..r,squarebinexp2lists(n)); if(0 <> AM_contains_zero_columns(am)) then RETURN(0); else RETURN(1); fi; end; A090064 := proc(n) option remember; local i; if(0 = n) then i := n; else i := A090064(n-1)+1; fi; while(i >= 0) # i.e. indefinitely, until the next one is found. do if(0 <> A090063(i)) then RETURN(i); fi; i := i+1; od; end; # Characteristic function for the next one: A090065 := proc(n) local r,am; r := sqrt(A000523(n)); if((not type(r,integer))) then RETURN(0); fi; am := array(1..r,1..r,squarebinexp2lists(n)); if(0 <> AM_contains_zero_columns_or_rows(am)) then RETURN(0); else RETURN(1); fi; end; A090061v2 := n -> A090063(A090060(n)); A090063v2 := n -> A090061(A090060(n)); A090065v2 := n -> A090061(n)*A090063(n); # Intersect of A090062 and A090064: A090066 := proc(n) option remember; local i; if(0 = n) then i := n; else i := A090066(n-1)+1; fi; while(i >= 0) # i.e. indefinitely, until the next one is found. do if(0 <> A090065(i)) then RETURN(i); fi; i := i+1; od; end; A090067 := n -> `if`((A000120(n) < 2),0,AMcertificate1(HG2AM(IB2HG(n)))); A090068 := n -> A090067(A090001(n)); A090069 := n -> A090067(A090011(n)); # Characteristic function for the next one: A090070 := n -> `if`((A000120(n) < 2),0,`if`((0 <> AM_contains_zero_columns_or_rows(HG2AM(IB2HG(n)))),0,1)); A090070v2 := n -> A090065(A090067(n)); # This defines contiguity: A090071 := proc(n) option remember; local i; if(0 = n) then i := n; else i := A090071(n-1)+1; fi; while(i >= 0) # i.e. indefinitely, until the next one is found. do if(0 <> A090070(i)) then RETURN(i); fi; i := i+1; od; end; # Continue from this idea: convert(divisors(27),list); --> [1,3,9,27] # [seq(IsGammoidOrGammier(convert(divisors(3^j),list),0,0),j=2..15)]; # [[3, 5, 2, 3], # [4, 7, 2, 5], # [5, 9, 2, 5], # [6, 11, 2, 7], # [7, 13, 2, 7], # [8, 15, 2, 12], # [9, 17, 2, 12], # [10, 19, 2, 12], # [11, 21, 2, 12], # [12, 23, 2, 12], # [13, 25, 2, 17], # [14, 27, 2, 17], # [15, 29, 2, 17], # [16, 31, 2, 17]] # Try also other bases, like 5^j, etc. ############################################################################## # # # Few example harmoids, gammoids, gammiers, for testing: # # # ############################################################################## Harmoid63 := IB2HG(63); # --> [1, 3, 5, 7, 9, 11] (fails congruity and fertility) Zarlino := IB2HG(151); # --> [1, 3, 5, 9, 15] ZarlinoSans3 := IB2HG(149); # --> [1, 5, 9, 15] ArabicZarlino := IB2HG(183); # --> [1, 3, 5, 9, 11, 15] (The gammier Arabic + Zarlino (1 9 5 11 3 15)) Gammoid5381 := IB2HG(5381); # --> [1, 5, 17, 21, 25] (not fertile) Harmoid139315 := IB2HG(139315); # --> [1, 3, 9, 11, 27, 35] (not congruous) PresqueGammier1183 := IB2HG(1183); # --> [1, 3, 5, 7, 9, 15, 21] DecatonicErlich := IB2HG(4195487); # --> [1, 3, 5, 7, 9, 15, 21, 45] # All three have 288 = 2^5 * 3^2 as their minimal example of divisor lattice: P5 := IB2HG(2658455991569831745807614120560697363); # --> [1, 3, 9, 27, 243] P6 := IB2HG(37576681324381331646231689548629392438010920782533117931316655544515344401833735095419183974156300348022587411); # --> [1, 3, 9, 81, 729] P6v2 := IB2HG(37576681324381331646231689548629392438010920782533117931316655544515344401833735095419183974156300348022595603); # --> [1, 3, 9, 27, 81, 729] results a duplicate Gammier. # Non-contiguous: NC125Harmoid := IB2HG(4611686018427392007); # --> [1, 3, 5, 25, 125] # AllAtoms(HG2Intervals(IB2HG(4611686018427392007))); --> [128/125, 25/24, 6/5] Some_harmoids := [Harmoid63,Zarlino,ZarlinoSans3,ArabicZarlino, Gammoid5381,Harmoid139315,PresqueGammier1183,DecatonicErlich, NC125Harmoid,P5,P6]; # [seq(IsGammoidOrGammier(Some_harmoids[i],0,0),i=1..nops(Some_harmoids))]; # --> [not congruous, [5, 13, 3, 7, 16], not contiguous, [6, 23, 5, 7, 176], # [5, 19, 5, 5, 64], not congruous, not contiguous,[8, 31, 4, 10, 472], # not contiguous, [5, 11, 2, 7, 9], [5, 13, 2, 7, 15]] output_seqs := proc(upto_n); read `c:\\maplev4\\matikka\\format.txt`; format([seq(A090042(n),n=0..upto_n)],90042); printf(`\n`); format([seq(A090043(n),n=0..upto_n)],90043); printf(`\n`); format([seq(A090044(n),n=0..upto_n)],90044); printf(`\n`); format([seq(A090045(n),n=0..upto_n)],90045); printf(`\n`); format([seq(A090046(n),n=0..upto_n)],90046); printf(`\n`); format([seq(A090047(n),n=0..upto_n)],90047); printf(`\n`); format([seq(A090048(n),n=1..upto_n)],90048); printf(`\n`); format([seq(A090049(n),n=1..upto_n)],90049); printf(`\n`); format([seq(A090050(n),n=1..upto_n)],90050); printf(`\n`); format([seq(A090051(n),n=1..upto_n)],90051); printf(`\n`); format([seq(A090052(n),n=1..upto_n)],90052); printf(`\n`); format([seq(A090053(n),n=1..upto_n)],90053); printf(`\n`); format([seq(A090054(n),n=0..upto_n)],90054); printf(`\n`); format([seq(A090055(n),n=0..upto_n)],90055); printf(`\n`); format([seq(A090056(n),n=0..upto_n)],90056); printf(`\n`); format([seq(A090057(n),n=0..upto_n)],90057); printf(`\n`); format([seq(A090058(n),n=0..upto_n)],90058); printf(`\n`); format([seq(A090059(n),n=0..upto_n)],90059); printf(`\n`); format([seq(A090060(n),n=0..upto_n)],90060); printf(`\n`); format([seq(A090061(n),n=0..upto_n)],90061); printf(`\n`); format([seq(A090062(n),n=0..upto_n)],90062); printf(`\n`); format([seq(A090063(n),n=0..upto_n)],90063); printf(`\n`); format([seq(A090064(n),n=0..upto_n)],90064); printf(`\n`); format([seq(A090065(n),n=0..upto_n)],90065); printf(`\n`); format([seq(A090066(n),n=0..upto_n)],90066); printf(`\n`); format([seq(A090067(n),n=0..upto_n)],90067); printf(`\n`); format([seq(A090068(n),n=1..upto_n)],90068); printf(`\n`); format([seq(A090069(n),n=1..upto_n)],90069); printf(`\n`); format([seq(A090070(n),n=0..upto_n)],90070); printf(`\n`); format([seq(A090071(n),n=0..upto_n)],90071); printf(`\n`); format(map(op,[seq(IB2HG(n),n=0..85)]),90072); printf(`\n`); format([seq(A090000(n),n=1..upto_n)],90000); printf(`\n`); format([seq(A090001(n),n=1..upto_n)],90001); printf(`\n`); format([seq(A090002(n),n=1..upto_n)],90002); printf(`\n`); format([seq(A090003(n),n=1..upto_n)],90003); printf(`\n`); format([seq(A090004(n),n=1..upto_n)],90004); printf(`\n`); format([seq(A090005(n),n=1..upto_n)],90005); printf(`\n`); format([seq(A090006(n),n=1..upto_n)],90006); printf(`\n`); format([seq(A090007(n),n=1..upto_n)],90007); printf(`\n`); format([seq(A090008(n),n=1..60)],90008); printf(`\n`); format([seq(A090009(n),n=1..60)],90009); printf(`\n`); format([seq(A090010(n),n=1..upto_n)],90010); printf(`\n`); format([seq(A090011(n),n=1..upto_n)],90011); printf(`\n`); format([seq(A090012(n),n=1..upto_n)],90012); printf(`\n`); format([seq(A090013(n),n=1..upto_n)],90013); printf(`\n`); format([seq(A090014(n),n=1..upto_n)],90014); printf(`\n`); format([seq(A090015(n),n=1..upto_n)],90015); printf(`\n`); format([seq(A090016(n),n=1..upto_n)],90016); printf(`\n`); format([seq(A090017(n),n=1..upto_n)],90017); printf(`\n`); format([seq(A090018(n),n=1..60)],90018); printf(`\n`); format([seq(A090019(n),n=1..60)],90019); printf(`\n`); format([seq(A090020(n),n=1..66)],90020); printf(`\n`); format([seq(A090021(n),n=1..66)],90021); printf(`\n`); # From pentatonic to enneatonic gammiers, positions + IB's: format([seq(A090022(n),n=1..30)],90022); printf(`\n`); # 5 format([seq(A090023(n),n=1..30)],90023); printf(`\n`); format([seq(A090024(n),n=1..100)],90024); printf(`\n`); # 6 format([seq(A090025(n),n=1..100)],90025); printf(`\n`); format([seq(A090026(n),n=1..100)],90026); printf(`\n`); # 7 format([seq(A090027(n),n=1..100)],90027); printf(`\n`); format([seq(A090028(n),n=1..100)],90028); printf(`\n`); # 8 format([seq(A090029(n),n=1..100)],90029); printf(`\n`); format([seq(A090030(n),n=1..100)],90030); printf(`\n`); # 9 format([seq(A090031(n),n=1..100)],90031); printf(`\n`); format([seq(A090032(n),n=1..100)],90032); printf(`\n`); # 10 format([seq(A090033(n),n=1..100)],90033); printf(`\n`); format([seq(A090034(n),n=1..100)],90034); printf(`\n`); # 11 format([seq(A090035(n),n=1..100)],90035); printf(`\n`); format([seq(A090036(n),n=1..100)],90036); printf(`\n`); # 12 format([seq(A090037(n),n=1..100)],90037); printf(`\n`); end; # The following list has been computed with # FindGammoids_or_Gammiers(16384,1,1) # Among these 1696 first Gammiers, the frequencies for gammiers # with n degrees (starting from n=5) are: # # 5 6 7 8 9 10 11 12 13 14 15 16 17 # 30, 113, 252, 354, 336, 263, 104, 200, 14, 19, 10, 0, 1 (<- IB 1633 for that loner) # # n A090011(n) [A090012(n),A090013(n),A090014(n),A090015(n)] # 1: ib 23: [4, 11, 3, 5] # 2: ib 27: [4, 11, 3, 5] # 3: ib 51: [4, 11, 3, 5] # 4: ib 55: [5, 19, 4, 7] # 5: ib 83: [4, 11, 3, 5] # 6: ib 91: [5, 19, 4, 7] # 7: ib 143: [5, 17, 4, 6] # 8: ib 147: [4, 11, 3, 5] # 9: ib 151: [5, 13, 3, 7] # 10: ib 155: [5, 19, 4, 7] # 11: ib 159: [6, 23, 5, 7] # 12: ib 175: [6, 27, 5, 7] # 13: ib 182: [5, 17, 4, 6] # 14: ib 183: [6, 23, 5, 7] # 15: ib 190: [6, 27, 5, 7] # 16: ib 199: [5, 17, 4, 6] # 17: ib 207: [6, 27, 5, 8] # 18: ib 211: [5, 19, 4, 7] # 19: ib 215: [6, 23, 5, 7] # 20: ib 219: [6, 29, 5, 9] # 21: ib 231: [6, 27, 5, 7] # 22: ib 239: [7, 39, 6, 9] # 23: ib 246: [6, 27, 5, 7] # 24: ib 254: [7, 39, 6, 8] # 25: ib 279: [5, 19, 4, 6] # 26: ib 283: [5, 19, 4, 6] # 27: ib 307: [5, 19, 4, 6] # 28: ib 311: [6, 29, 5, 8] # 29: ib 339: [5, 19, 4, 6] # 30: ib 347: [6, 29, 5, 8] # 31: ib 391: [5, 17, 4, 6] # 32: ib 403: [5, 19, 4, 6] # 33: ib 406: [5, 17, 4, 6] # 34: ib 407: [6, 23, 4, 10] # 35: ib 411: [6, 29, 5, 8] # 36: ib 414: [6, 27, 5, 7] # 37: ib 423: [6, 27, 5, 7] # 38: ib 467: [6, 29, 5, 8] # 39: ib 470: [6, 27, 5, 7] # 40: ib 475: [7, 41, 6, 10] # 41: ib 478: [7, 39, 6, 8] # 42: ib 479: [8, 49, 7, 10] # 43: ib 531: [4, 11, 3, 5] # 44: ib 535: [5, 19, 4, 7] # 45: ib 563: [5, 19, 4, 7] # 46: ib 567: [6, 29, 5, 9] # 47: ib 647: [5, 17, 4, 6] # 48: ib 663: [6, 23, 4, 9] # 49: ib 679: [6, 27, 5, 7] # 50: ib 694: [6, 27, 5, 7] # 51: ib 695: [7, 35, 6, 9] # 52: ib 702: [7, 39, 6, 8] # 53: ib 735: [8, 49, 7, 9] # 54: ib 758: [7, 39, 6, 8] # 55: ib 766: [8, 53, 7, 9] # 56: ib 787: [5, 19, 4, 6] # 57: ib 791: [6, 29, 5, 8] # 58: ib 819: [6, 29, 5, 8] # 59: ib 823: [7, 41, 6, 10] # 60: ib 903: [6, 27, 5, 8] # 61: ib 918: [6, 27, 5, 7] # 62: ib 919: [7, 35, 5, 12] # 63: ib 926: [7, 39, 6, 8] # 64: ib 935: [7, 39, 6, 9] # 65: ib 982: [7, 39, 6, 8] # 66: ib 990: [8, 53, 7, 9] # 67: ib 1043: [4, 11, 3, 5] # 68: ib 1047: [5, 19, 4, 7] # 69: ib 1054: [5, 17, 4, 6] # 70: ib 1055: [6, 23, 4, 7] # 71: ib 1067: [5, 17, 4, 6] # 72: ib 1071: [6, 27, 5, 7] # 73: ib 1075: [5, 19, 4, 7] # 74: ib 1079: [6, 29, 5, 9] # 75: ib 1083: [6, 23, 4, 7] # 76: ib 1086: [6, 27, 5, 7] # 77: ib 1087: [7, 35, 5, 9] # 78: ib 1114: [5, 17, 4, 6] # 79: ib 1115: [6, 23, 4, 7] # 80: ib 1131: [6, 27, 5, 7] # 81: ib 1135: [7, 39, 6, 8] # 82: ib 1146: [6, 27, 5, 7] # 83: ib 1151: [8, 49, 7, 9] # 84: ib 1163: [5, 17, 4, 6] # 85: ib 1165: [5, 17, 4, 6] # 86: ib 1166: [5, 17, 4, 6] # 87: ib 1174: [5, 17, 4, 6] # 88: ib 1175: [6, 23, 5, 7] # 89: ib 1179: [6, 23, 4, 7] # 90: ib 1180: [5, 17, 4, 6] # 91: ib 1196: [5, 17, 4, 6] # 92: ib 1198: [6, 27, 5, 8] # 93: ib 1199: [7, 33, 6, 8] # 94: ib 1206: [6, 27, 5, 8] # 95: ib 1223: [6, 27, 5, 7] # 96: ib 1227: [6, 27, 5, 7] # 97: ib 1228: [5, 17, 4, 6] # 98: ib 1230: [6, 27, 5, 8] # 99: ib 1238: [6, 27, 5, 7] # 100: ib 1242: [6, 27, 5, 7] # 101: ib 1243: [7, 35, 5, 9] # 102: ib 1246: [7, 33, 6, 8] # 103: ib 1255: [7, 39, 6, 8] # 104: ib 1260: [6, 27, 5, 8] # 105: ib 1262: [7, 39, 6, 10] # 106: ib 1263: [8, 47, 7, 10] # 107: ib 1270: [7, 39, 6, 9] # 108: ib 1274: [7, 39, 6, 8] # 109: ib 1275: [8, 49, 7, 9] # 110: ib 1278: [8, 47, 7, 10] # 111: ib 1299: [5, 19, 4, 6] # 112: ib 1303: [6, 29, 5, 8] # 113: ib 1310: [6, 27, 5, 7] # 114: ib 1311: [7, 35, 5, 8] # 115: ib 1323: [6, 27, 5, 7] # 116: ib 1327: [7, 39, 6, 8] # 117: ib 1331: [6, 29, 5, 8] # 118: ib 1335: [7, 41, 6, 10] # 119: ib 1339: [7, 35, 5, 8] # 120: ib 1342: [7, 39, 6, 8] # 121: ib 1343: [8, 49, 6, 10] # 122: ib 1370: [6, 27, 5, 7] # 123: ib 1371: [7, 35, 5, 8] # 124: ib 1387: [7, 39, 6, 8] # 125: ib 1391: [8, 53, 7, 9] # 126: ib 1402: [7, 39, 6, 8] # 127: ib 1407: [9, 65, 8, 10] # 128: ib 1415: [6, 27, 5, 7] # 129: ib 1419: [6, 27, 5, 7] # 130: ib 1420: [5, 17, 4, 6] # 131: ib 1421: [6, 27, 5, 8] # 132: ib 1423: [7, 33, 6, 8] # 133: ib 1435: [7, 35, 5, 8] # 134: ib 1436: [6, 27, 5, 8] # 135: ib 1437: [7, 39, 6, 10] # 136: ib 1438: [7, 33, 6, 8] # 137: ib 1439: [8, 41, 6, 10] # 138: ib 1447: [7, 39, 6, 8] # 139: ib 1463: [8, 49, 7, 10] # 140: ib 1471: [9, 57, 8, 10] # 141: ib 1483: [7, 39, 6, 8] # 142: ib 1498: [7, 39, 6, 8] # 143: ib 1499: [8, 49, 6, 10] # 144: ib 1503: [9, 57, 8, 10] # 145: ib 1530: [8, 53, 7, 9] # 146: ib 1531: [9, 65, 8, 10] # 147: ib 1555: [5, 19, 4, 7] # 148: ib 1559: [6, 29, 5, 9] # 149: ib 1562: [5, 17, 4, 6] # 150: ib 1563: [6, 23, 4, 7] # 151: ib 1566: [6, 27, 5, 8] # 152: ib 1567: [7, 35, 5, 9] # 153: ib 1579: [6, 27, 5, 7] # 154: ib 1583: [7, 39, 6, 8] # 155: ib 1587: [6, 29, 5, 9] # 156: ib 1591: [7, 41, 6, 11] # 157: ib 1594: [6, 27, 5, 7] # 158: ib 1595: [7, 35, 5, 9] # 159: ib 1598: [7, 39, 6, 9] # 160: ib 1599: [8, 49, 6, 11] # 161: ib 1643: [7, 39, 6, 8] # 162: ib 1647: [8, 53, 7, 9] # 163: ib 1659: [8, 49, 7, 9] # 164: ib 1671: [6, 27, 5, 7] # 165: ib 1675: [6, 27, 5, 7] # 166: ib 1676: [5, 17, 4, 6] # 167: ib 1677: [6, 27, 5, 8] # 168: ib 1679: [7, 33, 6, 8] # 169: ib 1686: [6, 27, 5, 7] # 170: ib 1687: [7, 35, 6, 9] # 171: ib 1690: [6, 27, 5, 7] # 172: ib 1691: [7, 35, 5, 9] # 173: ib 1692: [6, 27, 5, 8] # 174: ib 1703: [7, 39, 6, 8] # 175: ib 1718: [7, 39, 6, 9] # 176: ib 1722: [7, 39, 6, 8] # 177: ib 1723: [8, 49, 7, 9] # 178: ib 1739: [7, 39, 6, 8] # 179: ib 1750: [7, 39, 6, 8] # 180: ib 1755: [8, 49, 7, 9] # 181: ib 1782: [8, 53, 7, 10] # 182: ib 1811: [6, 29, 5, 8] # 183: ib 1815: [7, 41, 6, 10] # 184: ib 1818: [6, 27, 5, 7] # 185: ib 1819: [7, 35, 5, 8] # 186: ib 1822: [7, 39, 6, 9] # 187: ib 1823: [8, 49, 6, 10] # 188: ib 1835: [7, 39, 6, 8] # 189: ib 1839: [8, 53, 7, 9] # 190: ib 1843: [7, 41, 6, 10] # 191: ib 1847: [8, 55, 7, 12] # 192: ib 1850: [7, 39, 6, 8] # 193: ib 1851: [8, 49, 6, 10] # 194: ib 1854: [8, 53, 7, 10] # 195: ib 1855: [9, 65, 7, 12] # 196: ib 1899: [8, 53, 7, 9] # 197: ib 1903: [9, 69, 8, 10] # 198: ib 1915: [9, 65, 8, 10] # 199: ib 1927: [7, 39, 6, 9] # 200: ib 1931: [7, 39, 6, 8] # 201: ib 1932: [6, 27, 5, 8] # 202: ib 1933: [7, 39, 6, 10] # 203: ib 1946: [7, 39, 6, 8] # 204: ib 1947: [8, 49, 6, 10] # 205: ib 1948: [7, 39, 6, 10] # 206: ib 1949: [8, 53, 7, 12] # 207: ib 1950: [8, 47, 7, 10] # 208: ib 1951: [9, 57, 7, 12] # 209: ib 1959: [8, 53, 7, 10] # 210: ib 1975: [9, 65, 8, 12] # 211: ib 1978: [8, 53, 7, 9] # 212: ib 1979: [9, 65, 8, 10] # 213: ib 1983: [10, 75, 9, 12] # 214: ib 1995: [8, 53, 7, 9] # 215: ib 1999: [9, 63, 8, 10] # 216: ib 2011: [9, 65, 8, 10] # 217: ib 2067: [4, 11, 3, 5] # 218: ib 2071: [5, 19, 4, 7] # 219: ib 2099: [5, 19, 4, 7] # 220: ib 2103: [6, 29, 5, 9] # 221: ib 2191: [6, 27, 5, 7] # 222: ib 2198: [5, 17, 4, 6] # 223: ib 2199: [6, 23, 5, 7] # 224: ib 2206: [6, 27, 5, 7] # 225: ib 2223: [7, 39, 6, 8] # 226: ib 2230: [6, 27, 5, 8] # 227: ib 2238: [7, 39, 6, 9] # 228: ib 2247: [6, 27, 5, 7] # 229: ib 2255: [7, 39, 6, 9] # 230: ib 2262: [6, 27, 5, 7] # 231: ib 2270: [7, 39, 6, 8] # 232: ib 2279: [7, 39, 6, 8] # 233: ib 2287: [8, 53, 7, 10] # 234: ib 2294: [7, 39, 6, 9] # 235: ib 2302: [8, 53, 7, 10] # 236: ib 2323: [5, 19, 4, 6] # 237: ib 2327: [6, 29, 5, 8] # 238: ib 2355: [6, 29, 5, 8] # 239: ib 2359: [7, 41, 6, 10] # 240: ib 2439: [6, 27, 5, 7] # 241: ib 2471: [7, 39, 6, 8] # 242: ib 2487: [8, 49, 7, 10] # 243: ib 2579: [5, 19, 4, 7] # 244: ib 2583: [6, 29, 5, 9] # 245: ib 2611: [6, 29, 5, 9] # 246: ib 2615: [7, 41, 6, 11] # 247: ib 2695: [6, 27, 5, 7] # 248: ib 2710: [6, 27, 5, 7] # 249: ib 2711: [7, 35, 6, 9] # 250: ib 2718: [7, 39, 6, 8] # 251: ib 2727: [7, 39, 6, 8] # 252: ib 2742: [7, 39, 6, 9] # 253: ib 2750: [8, 53, 7, 10] # 254: ib 2774: [7, 39, 6, 8] # 255: ib 2782: [8, 53, 7, 9] # 256: ib 2806: [8, 53, 7, 10] # 257: ib 2814: [9, 69, 8, 11] # 258: ib 2835: [6, 29, 5, 8] # 259: ib 2839: [7, 41, 6, 10] # 260: ib 2867: [7, 41, 6, 10] # 261: ib 2871: [8, 55, 7, 12] # 262: ib 2951: [7, 39, 6, 9] # 263: ib 2983: [8, 53, 7, 10] # 264: ib 2999: [9, 65, 8, 12] # 265: ib 3083: [5, 17, 4, 6] # 266: ib 3087: [6, 27, 5, 7] # 267: ib 3091: [5, 19, 4, 7] # 268: ib 3095: [6, 29, 5, 9] # 269: ib 3099: [6, 23, 4, 7] # 270: ib 3102: [6, 27, 5, 7] # 271: ib 3103: [7, 35, 5, 9] # 272: ib 3115: [6, 27, 5, 8] # 273: ib 3119: [7, 39, 6, 9] # 274: ib 3123: [6, 29, 5, 9] # 275: ib 3127: [7, 41, 6, 11] # 276: ib 3131: [7, 35, 5, 9] # 277: ib 3134: [7, 39, 6, 8] # 278: ib 3135: [8, 49, 6, 11] # 279: ib 3147: [6, 27, 5, 7] # 280: ib 3151: [7, 39, 6, 8] # 281: ib 3162: [6, 27, 5, 7] # 282: ib 3167: [8, 49, 7, 9] # 283: ib 3179: [7, 39, 6, 9] # 284: ib 3183: [8, 53, 7, 10] # 285: ib 3194: [7, 39, 6, 8] # 286: ib 3199: [9, 65, 8, 11] # 287: ib 3212: [5, 17, 4, 6] # 288: ib 3214: [6, 27, 5, 8] # 289: ib 3215: [7, 33, 6, 8] # 290: ib 3222: [6, 27, 5, 8] # 291: ib 3244: [6, 27, 5, 8] # 292: ib 3246: [7, 39, 6, 10] # 293: ib 3254: [7, 39, 6, 10] # 294: ib 3271: [7, 39, 6, 8] # 295: ib 3276: [6, 27, 5, 8] # 296: ib 3278: [7, 39, 6, 10] # 297: ib 3279: [8, 47, 7, 10] # 298: ib 3286: [7, 39, 6, 9] # 299: ib 3290: [7, 39, 6, 8] # 300: ib 3291: [8, 49, 7, 9] # 301: ib 3294: [8, 47, 7, 10] # 302: ib 3303: [8, 53, 7, 9] # 303: ib 3308: [7, 39, 6, 10] # 304: ib 3310: [8, 53, 7, 12] # 305: ib 3318: [8, 53, 7, 11] # 306: ib 3322: [8, 53, 7, 9] # 307: ib 3326: [9, 63, 8, 12] # 308: ib 3339: [6, 27, 5, 7] # 309: ib 3343: [7, 39, 6, 8] # 310: ib 3347: [6, 29, 5, 8] # 311: ib 3351: [7, 41, 6, 10] # 312: ib 3355: [7, 35, 5, 8] # 313: ib 3358: [7, 39, 6, 8] # 314: ib 3359: [8, 49, 6, 10] # 315: ib 3371: [7, 39, 6, 9] # 316: ib 3375: [8, 53, 7, 10] # 317: ib 3379: [7, 41, 6, 10] # 318: ib 3383: [8, 55, 7, 12] # 319: ib 3387: [8, 49, 6, 10] # 320: ib 3390: [8, 53, 7, 9] # 321: ib 3391: [9, 65, 7, 12] # 322: ib 3403: [7, 39, 6, 8] # 323: ib 3407: [8, 53, 7, 9] # 324: ib 3418: [7, 39, 6, 8] # 325: ib 3423: [9, 65, 8, 10] # 326: ib 3435: [8, 53, 7, 10] # 327: ib 3439: [9, 69, 8, 11] # 328: ib 3450: [8, 53, 7, 9] # 329: ib 3455: [10, 83, 9, 12] # 330: ib 3463: [7, 39, 6, 8] # 331: ib 3479: [8, 49, 7, 10] # 332: ib 3487: [9, 57, 8, 10] # 333: ib 3495: [8, 53, 7, 9] # 334: ib 3546: [8, 53, 7, 9] # 335: ib 3547: [9, 65, 8, 10] # 336: ib 3578: [9, 69, 8, 10] # 337: ib 3595: [6, 27, 5, 7] # 338: ib 3599: [7, 39, 6, 8] # 339: ib 3603: [6, 29, 5, 9] # 340: ib 3607: [7, 41, 6, 11] # 341: ib 3610: [6, 27, 5, 7] # 342: ib 3611: [7, 35, 5, 9] # 343: ib 3614: [7, 39, 6, 9] # 344: ib 3615: [8, 49, 6, 11] # 345: ib 3627: [7, 39, 6, 9] # 346: ib 3631: [8, 53, 7, 10] # 347: ib 3635: [7, 41, 6, 11] # 348: ib 3639: [8, 55, 7, 13] # 349: ib 3642: [7, 39, 6, 8] # 350: ib 3643: [8, 49, 6, 11] # 351: ib 3646: [8, 53, 7, 10] # 352: ib 3647: [9, 65, 7, 13] # 353: ib 3659: [7, 39, 6, 8] # 354: ib 3663: [8, 53, 7, 9] # 355: ib 3675: [8, 49, 7, 9] # 356: ib 3691: [8, 53, 7, 10] # 357: ib 3695: [9, 69, 8, 11] # 358: ib 3707: [9, 65, 8, 11] # 359: ib 3719: [7, 39, 6, 8] # 360: ib 3734: [7, 39, 6, 9] # 361: ib 3738: [7, 39, 6, 8] # 362: ib 3739: [8, 49, 7, 9] # 363: ib 3751: [8, 53, 7, 9] # 364: ib 3766: [8, 53, 7, 11] # 365: ib 3770: [8, 53, 7, 9] # 366: ib 3798: [8, 53, 7, 10] # 367: ib 3830: [9, 69, 8, 12] # 368: ib 3851: [7, 39, 6, 8] # 369: ib 3855: [8, 53, 7, 9] # 370: ib 3859: [7, 41, 6, 10] # 371: ib 3863: [8, 55, 7, 12] # 372: ib 3866: [7, 39, 6, 8] # 373: ib 3867: [8, 49, 6, 10] # 374: ib 3870: [8, 53, 7, 10] # 375: ib 3871: [9, 65, 7, 12] # 376: ib 3883: [8, 53, 7, 10] # 377: ib 3887: [9, 69, 8, 11] # 378: ib 3891: [8, 55, 7, 12] # 379: ib 3895: [9, 71, 8, 14] # 380: ib 3898: [8, 53, 7, 9] # 381: ib 3899: [9, 65, 7, 12] # 382: ib 3902: [9, 69, 8, 11] # 383: ib 3903: [10, 83, 8, 14] # 384: ib 3915: [8, 53, 7, 9] # 385: ib 3919: [9, 69, 8, 10] # 386: ib 3931: [9, 65, 8, 10] # 387: ib 3947: [9, 69, 8, 11] # 388: ib 3951: [10, 87, 9, 12] # 389: ib 3963: [10, 83, 9, 12] # 390: ib 3975: [8, 53, 7, 10] # 391: ib 3991: [9, 65, 8, 12] # 392: ib 3994: [8, 53, 7, 9] # 393: ib 3995: [9, 65, 8, 10] # 394: ib 3999: [10, 75, 9, 12] # 395: ib 4007: [9, 69, 8, 11] # 396: ib 4026: [9, 69, 8, 10] # 397: ib 4103: [4, 11, 3, 5] # 398: ib 4111: [5, 19, 4, 6] # 399: ib 4115: [4, 11, 3, 5] # 400: ib 4117: [4, 11, 3, 5] # 401: ib 4123: [5, 19, 4, 7] # 402: ib 4125: [5, 19, 4, 6] # 403: ib 4133: [4, 11, 3, 5] # 404: ib 4135: [5, 19, 4, 7] # 405: ib 4141: [5, 19, 4, 6] # 406: ib 4143: [6, 29, 5, 8] # 407: ib 4167: [5, 19, 4, 6] # 408: ib 4175: [6, 29, 5, 7] # 409: ib 4179: [5, 19, 4, 7] # 410: ib 4181: [5, 19, 4, 6] # 411: ib 4187: [6, 29, 5, 9] # 412: ib 4189: [6, 29, 5, 7] # 413: ib 4197: [5, 19, 4, 6] # 414: ib 4199: [6, 29, 5, 8] # 415: ib 4205: [6, 29, 5, 7] # 416: ib 4207: [7, 41, 6, 9] # 417: ib 4231: [5, 13, 3, 6] # 418: ib 4238: [5, 17, 4, 6] # 419: ib 4239: [6, 23, 4, 9] # 420: ib 4241: [4, 11, 3, 5] # 421: ib 4248: [4, 11, 3, 5] # 422: ib 4250: [5, 19, 4, 6] # 423: ib 4251: [6, 27, 5, 9] # 424: ib 4252: [5, 19, 4, 6] # 425: ib 4253: [6, 27, 5, 7] # 426: ib 4261: [5, 19, 4, 6] # 427: ib 4262: [5, 17, 4, 6] # 428: ib 4269: [6, 29, 5, 7] # 429: ib 4271: [7, 35, 6, 9] # 430: ib 4273: [5, 19, 4, 6] # 431: ib 4277: [6, 27, 5, 7] # 432: ib 4280: [5, 19, 4, 6] # 433: ib 4282: [6, 29, 5, 7] # 434: ib 4284: [6, 29, 5, 7] # 435: ib 4286: [7, 35, 6, 8] # 436: ib 4294: [5, 17, 4, 6] # 437: ib 4295: [6, 23, 4, 9] # 438: ib 4302: [6, 27, 5, 8] # 439: ib 4303: [7, 35, 5, 12] # 440: ib 4304: [4, 11, 3, 5] # 441: ib 4306: [5, 19, 4, 6] # 442: ib 4307: [6, 27, 5, 9] # 443: ib 4308: [5, 19, 4, 6] # 444: ib 4309: [6, 27, 5, 7] # 445: ib 4312: [5, 19, 4, 7] # 446: ib 4314: [6, 29, 5, 8] # 447: ib 4316: [6, 29, 5, 8] # 448: ib 4325: [6, 29, 5, 7] # 449: ib 4327: [7, 35, 6, 9] # 450: ib 4333: [7, 41, 6, 8] # 451: ib 4336: [5, 19, 4, 6] # 452: ib 4338: [6, 29, 5, 7] # 453: ib 4340: [6, 29, 5, 7] # 454: ib 4342: [7, 35, 6, 8] # 455: ib 4344: [6, 29, 5, 8] # 456: ib 4346: [7, 41, 6, 9] # 457: ib 4348: [7, 41, 6, 9] # 458: ib 4357: [4, 11, 3, 5] # 459: ib 4365: [5, 19, 4, 6] # 460: ib 4371: [5, 19, 4, 6] # 461: ib 4373: [5, 19, 4, 7] # 462: ib 4379: [6, 29, 5, 8] # 463: ib 4381: [6, 29, 5, 8] # 464: ib 4421: [5, 19, 4, 6] # 465: ib 4429: [6, 29, 5, 7] # 466: ib 4435: [6, 29, 5, 8] # 467: ib 4437: [6, 29, 5, 8] # 468: ib 4443: [7, 41, 6, 10] # 469: ib 4445: [7, 41, 6, 9] # 470: ib 4485: [5, 19, 4, 6] # 471: ib 4493: [6, 29, 5, 7] # 472: ib 4494: [6, 27, 5, 7] # 473: ib 4496: [4, 11, 3, 5] # 474: ib 4497: [5, 19, 4, 7] # 475: ib 4498: [5, 19, 4, 6] # 476: ib 4500: [5, 19, 4, 6] # 477: ib 4510: [7, 35, 6, 8] # 478: ib 4518: [6, 27, 5, 7] # 479: ib 4528: [5, 19, 4, 6] # 480: ib 4529: [6, 29, 5, 8] # 481: ib 4530: [6, 29, 5, 7] # 482: ib 4532: [6, 29, 5, 7] # 483: ib 4534: [7, 35, 6, 8] # 484: ib 4549: [6, 29, 5, 7] # 485: ib 4550: [6, 27, 5, 7] # 486: ib 4557: [7, 41, 6, 8] # 487: ib 4558: [7, 39, 6, 9] # 488: ib 4566: [7, 35, 6, 8] # 489: ib 4571: [8, 53, 7, 12] # 490: ib 4573: [8, 53, 7, 10] # 491: ib 4613: [4, 11, 3, 5] # 492: ib 4621: [5, 19, 4, 6] # 493: ib 4629: [5, 19, 4, 7] # 494: ib 4637: [6, 29, 5, 8] # 495: ib 4677: [5, 19, 4, 6] # 496: ib 4685: [6, 29, 5, 7] # 497: ib 4693: [6, 29, 5, 8] # 498: ib 4701: [7, 41, 6, 9] # 499: ib 4741: [5, 19, 4, 6] # 500: ib 4749: [6, 29, 5, 7] # 501: ib 4750: [6, 27, 5, 7] # 502: ib 4753: [5, 19, 4, 6] # 503: ib 4760: [5, 19, 4, 6] # 504: ib 4762: [6, 29, 5, 7] # 505: ib 4764: [6, 29, 5, 7] # 506: ib 4765: [7, 39, 6, 9] # 507: ib 4767: [8, 45, 7, 9] # 508: ib 4774: [6, 27, 5, 7] # 509: ib 4785: [6, 29, 5, 7] # 510: ib 4791: [8, 45, 7, 9] # 511: ib 4792: [6, 29, 5, 7] # 512: ib 4794: [7, 41, 6, 8] # 513: ib 4796: [7, 41, 6, 8] # 514: ib 4798: [8, 49, 7, 9] # 515: ib 4805: [6, 29, 5, 7] # 516: ib 4806: [6, 27, 5, 7] # 517: ib 4813: [7, 41, 6, 8] # 518: ib 4814: [7, 39, 6, 9] # 519: ib 4816: [5, 19, 4, 6] # 520: ib 4818: [6, 29, 5, 7] # 521: ib 4820: [6, 29, 5, 7] # 522: ib 4821: [7, 39, 6, 9] # 523: ib 4823: [8, 45, 7, 9] # 524: ib 4824: [6, 29, 5, 8] # 525: ib 4826: [7, 41, 6, 9] # 526: ib 4828: [7, 41, 6, 9] # 527: ib 4848: [6, 29, 5, 7] # 528: ib 4850: [7, 41, 6, 8] # 529: ib 4852: [7, 41, 6, 8] # 530: ib 4854: [8, 49, 7, 9] # 531: ib 4856: [7, 41, 6, 9] # 532: ib 4858: [8, 55, 7, 10] # 533: ib 4860: [8, 55, 7, 10] # 534: ib 4869: [5, 19, 4, 7] # 535: ib 4877: [6, 29, 5, 8] # 536: ib 4885: [6, 29, 5, 9] # 537: ib 4893: [7, 41, 6, 10] # 538: ib 4933: [6, 29, 5, 8] # 539: ib 4941: [7, 41, 6, 9] # 540: ib 4949: [7, 41, 6, 10] # 541: ib 4957: [8, 55, 7, 11] # 542: ib 4997: [6, 29, 5, 8] # 543: ib 5005: [7, 41, 6, 9] # 544: ib 5006: [7, 39, 6, 8] # 545: ib 5007: [8, 49, 7, 9] # 546: ib 5008: [5, 19, 4, 6] # 547: ib 5009: [6, 29, 5, 8] # 548: ib 5010: [6, 29, 5, 7] # 549: ib 5012: [6, 29, 5, 7] # 550: ib 5022: [8, 49, 7, 9] # 551: ib 5030: [7, 39, 6, 8] # 552: ib 5031: [8, 49, 7, 9] # 553: ib 5040: [6, 29, 5, 7] # 554: ib 5041: [7, 41, 6, 9] # 555: ib 5042: [7, 41, 6, 8] # 556: ib 5044: [7, 41, 6, 8] # 557: ib 5046: [8, 49, 7, 9] # 558: ib 5061: [7, 41, 6, 9] # 559: ib 5062: [7, 39, 6, 8] # 560: ib 5063: [8, 49, 7, 9] # 561: ib 5069: [8, 55, 7, 10] # 562: ib 5070: [8, 53, 7, 10] # 563: ib 5078: [8, 49, 7, 9] # 564: ib 5085: [9, 69, 8, 12] # 565: ib 5087: [10, 79, 9, 12] # 566: ib 5125: [4, 11, 3, 5] # 567: ib 5127: [5, 19, 4, 7] # 568: ib 5133: [5, 19, 4, 6] # 569: ib 5135: [6, 25, 4, 8] # 570: ib 5146: [5, 17, 4, 6] # 571: ib 5147: [6, 23, 4, 7] # 572: ib 5157: [5, 19, 4, 7] # 573: ib 5159: [6, 29, 5, 9] # 574: ib 5163: [6, 27, 5, 7] # 575: ib 5165: [6, 29, 5, 8] # 576: ib 5178: [6, 27, 5, 7] # 577: ib 5189: [5, 19, 4, 6] # 578: ib 5191: [6, 29, 5, 8] # 579: ib 5197: [6, 29, 5, 7] # 580: ib 5199: [7, 37, 5, 9] # 581: ib 5210: [6, 27, 5, 8] # 582: ib 5211: [7, 35, 5, 9] # 583: ib 5221: [6, 29, 5, 8] # 584: ib 5223: [7, 41, 6, 10] # 585: ib 5227: [7, 39, 6, 8] # 586: ib 5229: [7, 41, 6, 9] # 587: ib 5242: [7, 39, 6, 9] # 588: ib 5253: [5, 19, 4, 6] # 589: ib 5254: [5, 17, 4, 6] # 590: ib 5260: [5, 17, 4, 6] # 591: ib 5262: [6, 23, 5, 8] # 592: ib 5265: [5, 19, 4, 6] # 593: ib 5269: [6, 27, 5, 7] # 594: ib 5272: [5, 19, 4, 6] # 595: ib 5275: [7, 33, 5, 9] # 596: ib 5278: [7, 29, 6, 8] # 597: ib 5285: [6, 29, 5, 8] # 598: ib 5286: [6, 27, 5, 8] # 599: ib 5292: [6, 27, 5, 8] # 600: ib 5297: [6, 29, 5, 7] # 601: ib 5304: [6, 29, 5, 7] # 602: ib 5307: [8, 47, 7, 9] # 603: ib 5308: [7, 37, 6, 8] # 604: ib 5317: [6, 29, 5, 7] # 605: ib 5319: [7, 35, 6, 9] # 606: ib 5324: [6, 27, 5, 8] # 607: ib 5325: [7, 37, 6, 8] # 608: ib 5328: [5, 19, 4, 6] # 609: ib 5330: [6, 29, 5, 7] # 610: ib 5332: [6, 29, 5, 7] # 611: ib 5334: [7, 35, 6, 8] # 612: ib 5336: [6, 29, 5, 8] # 613: ib 5349: [7, 41, 6, 9] # 614: ib 5356: [7, 39, 6, 10] # 615: ib 5357: [8, 51, 7, 10] # 616: ib 5358: [8, 49, 7, 12] # 617: ib 5359: [9, 59, 8, 12] # 618: ib 5360: [6, 29, 5, 7] # 619: ib 5362: [7, 41, 6, 8] # 620: ib 5364: [7, 41, 6, 8] # 621: ib 5368: [7, 41, 6, 9] # 622: ib 5372: [8, 51, 7, 10] # 623: ib 5402: [6, 27, 5, 7] # 624: ib 5403: [7, 35, 5, 8] # 625: ib 5419: [7, 39, 6, 8] # 626: ib 5434: [7, 39, 6, 8] # 627: ib 5466: [7, 39, 6, 9] # 628: ib 5467: [8, 49, 6, 10] # 629: ib 5483: [8, 53, 7, 9] # 630: ib 5498: [8, 53, 7, 10] # 631: ib 5510: [6, 27, 5, 7] # 632: ib 5520: [5, 19, 4, 6] # 633: ib 5521: [6, 29, 5, 8] # 634: ib 5522: [6, 29, 5, 7] # 635: ib 5524: [6, 29, 5, 7] # 636: ib 5526: [7, 35, 6, 8] # 637: ib 5530: [7, 37, 6, 8] # 638: ib 5542: [7, 39, 6, 9] # 639: ib 5552: [6, 29, 5, 7] # 640: ib 5553: [7, 41, 6, 9] # 641: ib 5554: [7, 41, 6, 8] # 642: ib 5556: [7, 41, 6, 8] # 643: ib 5557: [8, 53, 7, 10] # 644: ib 5562: [8, 51, 7, 9] # 645: ib 5595: [9, 63, 7, 12] # 646: ib 5627: [10, 81, 9, 12] # 647: ib 5675: [7, 39, 6, 8] # 648: ib 5691: [8, 49, 7, 9] # 649: ib 5739: [8, 53, 7, 9] # 650: ib 5766: [6, 27, 5, 7] # 651: ib 5777: [6, 29, 5, 7] # 652: ib 5783: [8, 45, 7, 9] # 653: ib 5784: [6, 29, 5, 7] # 654: ib 5786: [7, 37, 6, 8] # 655: ib 5787: [8, 47, 7, 9] # 656: ib 5798: [7, 39, 6, 9] # 657: ib 5809: [7, 41, 6, 8] # 658: ib 5811: [8, 53, 7, 9] # 659: ib 5813: [8, 53, 7, 9] # 660: ib 5816: [7, 41, 6, 8] # 661: ib 5818: [8, 51, 7, 9] # 662: ib 5840: [6, 29, 5, 7] # 663: ib 5842: [7, 41, 6, 8] # 664: ib 5844: [7, 41, 6, 8] # 665: ib 5846: [8, 49, 7, 9] # 666: ib 5848: [7, 41, 6, 9] # 667: ib 5872: [7, 41, 6, 8] # 668: ib 5874: [8, 55, 7, 9] # 669: ib 5876: [8, 55, 7, 9] # 670: ib 5880: [8, 55, 7, 10] # 671: ib 5886: [10, 77, 9, 12] # 672: ib 5931: [8, 53, 7, 9] # 673: ib 5947: [9, 65, 8, 10] # 674: ib 5995: [9, 69, 8, 10] # 675: ib 6015: [11, 101, 10, 12] # 676: ib 6022: [7, 39, 6, 8] # 677: ib 6023: [8, 49, 7, 9] # 678: ib 6027: [8, 53, 7, 9] # 679: ib 6032: [6, 29, 5, 7] # 680: ib 6033: [7, 41, 6, 9] # 681: ib 6034: [7, 41, 6, 8] # 682: ib 6036: [7, 41, 6, 8] # 683: ib 6038: [8, 49, 7, 9] # 684: ib 6054: [8, 53, 7, 10] # 685: ib 6064: [7, 41, 6, 8] # 686: ib 6065: [8, 55, 7, 10] # 687: ib 6066: [8, 55, 7, 9] # 688: ib 6068: [8, 55, 7, 9] # 689: ib 6071: [10, 79, 9, 12] # 690: ib 6111: [11, 91, 10, 12] # 691: ib 6149: [4, 11, 3, 5] # 692: ib 6151: [5, 19, 4, 7] # 693: ib 6157: [5, 19, 4, 6] # 694: ib 6159: [6, 29, 5, 8] # 695: ib 6181: [5, 19, 4, 7] # 696: ib 6183: [6, 29, 5, 9] # 697: ib 6189: [6, 29, 5, 8] # 698: ib 6191: [7, 41, 6, 10] # 699: ib 6213: [5, 19, 4, 6] # 700: ib 6215: [6, 29, 5, 8] # 701: ib 6221: [6, 29, 5, 7] # 702: ib 6223: [7, 41, 6, 9] # 703: ib 6245: [6, 29, 5, 8] # 704: ib 6247: [7, 41, 6, 10] # 705: ib 6253: [7, 41, 6, 9] # 706: ib 6255: [8, 55, 7, 11] # 707: ib 6277: [5, 19, 4, 6] # 708: ib 6278: [5, 17, 4, 6] # 709: ib 6285: [6, 29, 5, 7] # 710: ib 6287: [7, 35, 6, 9] # 711: ib 6289: [5, 19, 4, 6] # 712: ib 6293: [6, 27, 5, 7] # 713: ib 6296: [5, 19, 4, 6] # 714: ib 6298: [6, 29, 5, 7] # 715: ib 6300: [6, 29, 5, 7] # 716: ib 6302: [7, 35, 6, 8] # 717: ib 6309: [6, 29, 5, 8] # 718: ib 6310: [6, 27, 5, 8] # 719: ib 6317: [7, 41, 6, 9] # 720: ib 6321: [6, 29, 5, 7] # 721: ib 6328: [6, 29, 5, 7] # 722: ib 6330: [7, 41, 6, 8] # 723: ib 6332: [7, 41, 6, 8] # 724: ib 6341: [6, 29, 5, 7] # 725: ib 6343: [7, 35, 6, 9] # 726: ib 6349: [7, 41, 6, 8] # 727: ib 6352: [5, 19, 4, 6] # 728: ib 6354: [6, 29, 5, 7] # 729: ib 6356: [6, 29, 5, 7] # 730: ib 6358: [7, 35, 6, 8] # 731: ib 6360: [6, 29, 5, 8] # 732: ib 6362: [7, 41, 6, 9] # 733: ib 6364: [7, 41, 6, 9] # 734: ib 6373: [7, 41, 6, 9] # 735: ib 6381: [8, 55, 7, 10] # 736: ib 6383: [9, 65, 8, 12] # 737: ib 6384: [6, 29, 5, 7] # 738: ib 6386: [7, 41, 6, 8] # 739: ib 6388: [7, 41, 6, 8] # 740: ib 6392: [7, 41, 6, 9] # 741: ib 6394: [8, 55, 7, 10] # 742: ib 6396: [8, 55, 7, 10] # 743: ib 6398: [9, 65, 8, 11] # 744: ib 6534: [6, 27, 5, 7] # 745: ib 6544: [5, 19, 4, 6] # 746: ib 6545: [6, 29, 5, 8] # 747: ib 6546: [6, 29, 5, 7] # 748: ib 6548: [6, 29, 5, 7] # 749: ib 6550: [7, 35, 6, 8] # 750: ib 6566: [7, 39, 6, 9] # 751: ib 6576: [6, 29, 5, 7] # 752: ib 6577: [7, 41, 6, 9] # 753: ib 6578: [7, 41, 6, 8] # 754: ib 6580: [7, 41, 6, 8] # 755: ib 6581: [8, 53, 7, 10] # 756: ib 6790: [6, 27, 5, 7] # 757: ib 6801: [6, 29, 5, 7] # 758: ib 6807: [8, 45, 7, 9] # 759: ib 6808: [6, 29, 5, 7] # 760: ib 6810: [7, 41, 6, 8] # 761: ib 6812: [7, 41, 6, 8] # 762: ib 6814: [8, 49, 7, 9] # 763: ib 6822: [7, 39, 6, 9] # 764: ib 6833: [7, 41, 6, 8] # 765: ib 6835: [8, 53, 7, 9] # 766: ib 6837: [8, 53, 7, 9] # 767: ib 6840: [7, 41, 6, 8] # 768: ib 6842: [8, 55, 7, 9] # 769: ib 6844: [8, 55, 7, 9] # 770: ib 6864: [6, 29, 5, 7] # 771: ib 6866: [7, 41, 6, 8] # 772: ib 6868: [7, 41, 6, 8] # 773: ib 6870: [8, 49, 7, 9] # 774: ib 6872: [7, 41, 6, 9] # 775: ib 6874: [8, 55, 7, 10] # 776: ib 6876: [8, 55, 7, 10] # 777: ib 6896: [7, 41, 6, 8] # 778: ib 6898: [8, 55, 7, 9] # 779: ib 6900: [8, 55, 7, 9] # 780: ib 6904: [8, 55, 7, 10] # 781: ib 6906: [9, 71, 8, 11] # 782: ib 6908: [9, 71, 8, 11] # 783: ib 6910: [10, 83, 9, 12] # 784: ib 7046: [7, 39, 6, 8] # 785: ib 7047: [8, 49, 7, 9] # 786: ib 7056: [6, 29, 5, 7] # 787: ib 7057: [7, 41, 6, 9] # 788: ib 7058: [7, 41, 6, 8] # 789: ib 7060: [7, 41, 6, 8] # 790: ib 7062: [8, 49, 7, 9] # 791: ib 7078: [8, 53, 7, 10] # 792: ib 7088: [7, 41, 6, 8] # 793: ib 7089: [8, 55, 7, 10] # 794: ib 7090: [8, 55, 7, 9] # 795: ib 7092: [8, 55, 7, 9] # 796: ib 7095: [10, 79, 9, 12] # 797: ib 7173: [5, 19, 4, 7] # 798: ib 7175: [6, 29, 5, 9] # 799: ib 7179: [6, 27, 5, 7] # 800: ib 7181: [6, 29, 5, 8] # 801: ib 7194: [6, 27, 5, 7] # 802: ib 7205: [6, 29, 5, 9] # 803: ib 7207: [7, 41, 6, 11] # 804: ib 7211: [7, 39, 6, 9] # 805: ib 7213: [7, 41, 6, 10] # 806: ib 7226: [7, 39, 6, 8] # 807: ib 7237: [6, 29, 5, 8] # 808: ib 7239: [7, 41, 6, 10] # 809: ib 7243: [7, 39, 6, 8] # 810: ib 7245: [7, 41, 6, 9] # 811: ib 7258: [7, 39, 6, 9] # 812: ib 7269: [7, 41, 6, 10] # 813: ib 7271: [8, 55, 7, 12] # 814: ib 7275: [8, 53, 7, 10] # 815: ib 7277: [8, 55, 7, 11] # 816: ib 7290: [8, 53, 7, 10] # 817: ib 7301: [6, 29, 5, 8] # 818: ib 7302: [6, 27, 5, 8] # 819: ib 7308: [6, 27, 5, 8] # 820: ib 7313: [6, 29, 5, 7] # 821: ib 7320: [6, 29, 5, 7] # 822: ib 7323: [8, 47, 7, 9] # 823: ib 7324: [7, 37, 6, 8] # 824: ib 7333: [7, 41, 6, 10] # 825: ib 7334: [7, 39, 6, 10] # 826: ib 7340: [7, 39, 6, 10] # 827: ib 7345: [7, 41, 6, 8] # 828: ib 7347: [8, 53, 7, 9] # 829: ib 7352: [7, 41, 6, 8] # 830: ib 7365: [7, 41, 6, 9] # 831: ib 7372: [7, 39, 6, 10] # 832: ib 7373: [8, 51, 7, 10] # 833: ib 7374: [8, 49, 7, 12] # 834: ib 7375: [9, 59, 8, 12] # 835: ib 7376: [6, 29, 5, 7] # 836: ib 7378: [7, 41, 6, 8] # 837: ib 7380: [7, 41, 6, 8] # 838: ib 7384: [7, 41, 6, 9] # 839: ib 7388: [8, 51, 7, 10] # 840: ib 7397: [8, 55, 7, 11] # 841: ib 7404: [8, 53, 7, 12] # 842: ib 7405: [9, 67, 8, 12] # 843: ib 7408: [7, 41, 6, 8] # 844: ib 7410: [8, 55, 7, 9] # 845: ib 7412: [8, 55, 7, 9] # 846: ib 7416: [8, 55, 7, 10] # 847: ib 7435: [7, 39, 6, 8] # 848: ib 7450: [7, 39, 6, 8] # 849: ib 7467: [8, 53, 7, 10] # 850: ib 7482: [8, 53, 7, 9] # 851: ib 7499: [8, 53, 7, 9] # 852: ib 7514: [8, 53, 7, 10] # 853: ib 7531: [9, 69, 8, 11] # 854: ib 7546: [9, 69, 8, 11] # 855: ib 7558: [7, 39, 6, 9] # 856: ib 7568: [6, 29, 5, 7] # 857: ib 7569: [7, 41, 6, 9] # 858: ib 7570: [7, 41, 6, 8] # 859: ib 7572: [7, 41, 6, 8] # 860: ib 7573: [8, 53, 7, 10] # 861: ib 7578: [8, 51, 7, 9] # 862: ib 7590: [8, 53, 7, 11] # 863: ib 7600: [7, 41, 6, 8] # 864: ib 7601: [8, 55, 7, 10] # 865: ib 7602: [8, 55, 7, 9] # 866: ib 7604: [8, 55, 7, 9] # 867: ib 7610: [9, 67, 8, 10] # 868: ib 7643: [10, 81, 9, 12] # 869: ib 7691: [7, 39, 6, 8] # 870: ib 7707: [8, 49, 7, 9] # 871: ib 7723: [8, 53, 7, 10] # 872: ib 7739: [9, 65, 8, 11] # 873: ib 7755: [8, 53, 7, 9] # 874: ib 7787: [9, 69, 8, 11] # 875: ib 7814: [7, 39, 6, 9] # 876: ib 7825: [7, 41, 6, 8] # 877: ib 7827: [8, 53, 7, 9] # 878: ib 7829: [8, 53, 7, 9] # 879: ib 7832: [7, 41, 6, 8] # 880: ib 7834: [8, 51, 7, 9] # 881: ib 7846: [8, 53, 7, 11] # 882: ib 7857: [8, 55, 7, 9] # 883: ib 7864: [8, 55, 7, 9] # 884: ib 7866: [9, 67, 8, 10] # 885: ib 7868: [9, 67, 8, 10] # 886: ib 7888: [7, 41, 6, 8] # 887: ib 7890: [8, 55, 7, 9] # 888: ib 7892: [8, 55, 7, 9] # 889: ib 7896: [8, 55, 7, 10] # 890: ib 7902: [10, 77, 9, 12] # 891: ib 7920: [8, 55, 7, 9] # 892: ib 7922: [9, 71, 8, 10] # 893: ib 7924: [9, 71, 8, 10] # 894: ib 7928: [9, 71, 8, 11] # 895: ib 7932: [10, 85, 9, 12] # 896: ib 7947: [8, 53, 7, 9] # 897: ib 7963: [9, 65, 8, 10] # 898: ib 7979: [9, 69, 8, 11] # 899: ib 7995: [10, 83, 9, 12] # 900: ib 8011: [9, 69, 8, 10] # 901: ib 8031: [11, 101, 10, 12] # 902: ib 8043: [10, 87, 9, 12] # 903: ib 8070: [8, 53, 7, 10] # 904: ib 8080: [7, 41, 6, 8] # 905: ib 8081: [8, 55, 7, 10] # 906: ib 8082: [8, 55, 7, 9] # 907: ib 8084: [8, 55, 7, 9] # 908: ib 8087: [10, 79, 9, 12] # 909: ib 8102: [9, 69, 8, 12] # 910: ib 8112: [8, 55, 7, 9] # 911: ib 8113: [9, 71, 8, 11] # 912: ib 8114: [9, 71, 8, 10] # 913: ib 8115: [10, 87, 9, 12] # 914: ib 8116: [9, 71, 8, 10] # 915: ib 8117: [10, 87, 9, 12] # 916: ib 8211: [4, 7, 2, 5] # 917: ib 8214: [4, 11, 3, 5] # 918: ib 8218: [4, 11, 3, 5] # 919: ib 8219: [5, 15, 3, 7] # 920: ib 8242: [4, 11, 3, 5] # 921: ib 8246: [5, 19, 4, 7] # 922: ib 8275: [5, 15, 3, 8] # 923: ib 8278: [5, 19, 4, 6] # 924: ib 8282: [5, 19, 4, 6] # 925: ib 8283: [6, 25, 4, 10] # 926: ib 8306: [5, 19, 4, 6] # 927: ib 8310: [6, 29, 5, 8] # 928: ib 8311: [7, 37, 6, 8] # 929: ib 8338: [4, 11, 3, 5], duplicate of A090011(1)=23 (Here HG=[3,9,15,27], while 23 has: HG=[1,3,5,9]) # 930: ib 8339: [5, 15, 3, 7] # 931: ib 8343: [6, 17, 3, 7] # 932: ib 8346: [5, 19, 4, 7] # 933: ib 8347: [6, 25, 4, 9] # 934: ib 8348: [5, 17, 4, 6] # 935: ib 8350: [6, 23, 4, 7] # 936: ib 8351: [7, 29, 4, 9] # 937: ib 8359: [6, 27, 5, 7] # 938: ib 8367: [7, 39, 6, 9] # 939: ib 8374: [6, 23, 4, 7] # 940: ib 8380: [6, 27, 5, 7] # 941: ib 8382: [7, 35, 5, 9] # 942: ib 8383: [8, 43, 6, 9] # 943: ib 8402: [5, 19, 4, 6] # 944: ib 8403: [6, 25, 4, 10] # 945: ib 8407: [7, 29, 4, 10] # 946: ib 8410: [6, 29, 5, 8] # 947: ib 8411: [7, 37, 5, 12] # 948: ib 8412: [6, 27, 5, 7] # 949: ib 8414: [7, 35, 5, 8] # 950: ib 8415: [8, 43, 5, 12] # 951: ib 8438: [7, 35, 5, 8] # 952: ib 8444: [7, 39, 6, 8] # 953: ib 8445: [8, 53, 7, 9] # 954: ib 8446: [8, 49, 6, 10] # 955: ib 8466: [4, 11, 3, 5] # 956: ib 8467: [5, 15, 3, 8] # 957: ib 8474: [5, 19, 4, 7] # 958: ib 8475: [6, 25, 4, 10] # 959: ib 8503: [7, 37, 6, 8] # 960: ib 8530: [5, 19, 4, 6] # 961: ib 8531: [6, 25, 5, 8] # 962: ib 8538: [6, 29, 5, 8] # 963: ib 8539: [7, 37, 6, 10] # 964: ib 8594: [5, 19, 4, 7] # 965: ib 8595: [6, 25, 4, 10] # 966: ib 8598: [6, 23, 4, 7] # 967: ib 8602: [6, 29, 5, 9] # 968: ib 8603: [7, 37, 5, 12] # 969: ib 8604: [6, 27, 5, 7] # 970: ib 8606: [7, 35, 5, 9] # 971: ib 8636: [7, 39, 6, 8] # 972: ib 8638: [8, 49, 7, 9] # 973: ib 8658: [6, 29, 5, 8] # 974: ib 8659: [7, 37, 6, 10] # 975: ib 8662: [7, 35, 5, 8] # 976: ib 8663: [8, 43, 6, 10] # 977: ib 8666: [7, 41, 6, 10] # 978: ib 8667: [8, 51, 7, 12] # 979: ib 8668: [7, 39, 6, 8] # 980: ib 8670: [8, 49, 6, 10] # 981: ib 8671: [9, 59, 7, 12] # 982: ib 8700: [8, 53, 7, 9] # 983: ib 8702: [9, 65, 8, 10] # 984: ib 8722: [4, 11, 3, 5] # 985: ib 8726: [5, 19, 4, 7] # 986: ib 8754: [5, 19, 4, 7] # 987: ib 8758: [6, 29, 5, 9] # 988: ib 8786: [5, 19, 4, 6] # 989: ib 8790: [6, 29, 5, 8] # 990: ib 8791: [7, 37, 6, 8] # 991: ib 8818: [6, 29, 5, 8] # 992: ib 8819: [7, 37, 6, 8] # 993: ib 8822: [7, 41, 6, 10] # 994: ib 8831: [9, 67, 8, 10] # 995: ib 8852: [5, 17, 4, 6] # 996: ib 8854: [6, 23, 4, 7] # 997: ib 8855: [7, 29, 4, 9] # 998: ib 8863: [8, 43, 6, 9] # 999: ib 8884: [6, 27, 5, 7] # 1000: ib 8886: [7, 35, 5, 9] # 1001: ib 8887: [8, 43, 6, 9] # 1002: ib 8894: [8, 49, 7, 9] # 1003: ib 8916: [6, 27, 5, 7] # 1004: ib 8918: [7, 35, 5, 8] # 1005: ib 8927: [9, 59, 7, 12] # 1006: ib 8948: [7, 39, 6, 8] # 1007: ib 8949: [8, 53, 7, 9] # 1008: ib 8950: [8, 49, 6, 10] # 1009: ib 8958: [9, 65, 8, 10] # 1010: ib 8983: [7, 37, 6, 8] # 1011: ib 9011: [7, 37, 6, 8] # 1012: ib 9023: [9, 67, 8, 10] # 1013: ib 9108: [6, 27, 5, 7] # 1014: ib 9118: [8, 49, 7, 9] # 1015: ib 9140: [7, 39, 6, 8] # 1016: ib 9142: [8, 49, 7, 9] # 1017: ib 9172: [7, 39, 6, 8] # 1018: ib 9175: [9, 59, 7, 12] # 1019: ib 9182: [9, 65, 8, 10] # 1020: ib 9183: [10, 77, 9, 12] # 1021: ib 9204: [8, 53, 7, 9] # 1022: ib 9206: [9, 65, 8, 10] # 1023: ib 9234: [4, 11, 3, 5], duplicate of A090011(2)=27 (Here HG=[3,9,21,27]), while 27 has: HG=[1,3,7,9]) # 1024: ib 9238: [5, 19, 4, 7] # 1025: ib 9242: [5, 13, 3, 7] # 1026: ib 9243: [6, 17, 3, 7] # 1027: ib 9245: [6, 27, 5, 7] # 1028: ib 9246: [6, 23, 5, 7] # 1029: ib 9266: [5, 19, 4, 7] # 1030: ib 9270: [6, 29, 5, 9] # 1031: ib 9272: [5, 17, 4, 6] # 1032: ib 9274: [6, 23, 4, 9] # 1033: ib 9275: [7, 29, 4, 9] # 1034: ib 9276: [6, 27, 5, 7] # 1035: ib 9278: [7, 35, 6, 9] # 1036: ib 9279: [8, 43, 6, 9] # 1037: ib 9298: [5, 19, 4, 6] # 1038: ib 9302: [6, 29, 5, 8] # 1039: ib 9303: [7, 37, 6, 8] # 1040: ib 9304: [5, 17, 4, 6] # 1041: ib 9306: [6, 23, 4, 10] # 1042: ib 9308: [6, 27, 5, 7] # 1043: ib 9330: [6, 29, 5, 8] # 1044: ib 9331: [7, 37, 6, 8] # 1045: ib 9334: [7, 41, 6, 10] # 1046: ib 9336: [6, 27, 5, 8] # 1047: ib 9338: [7, 35, 5, 12] # 1048: ib 9340: [7, 39, 6, 9] # 1049: ib 9351: [6, 27, 5, 7] # 1050: ib 9356: [5, 17, 4, 6] # 1051: ib 9366: [6, 23, 4, 7] # 1052: ib 9368: [5, 17, 4, 6] # 1053: ib 9370: [6, 23, 5, 7] # 1054: ib 9371: [7, 29, 4, 9] # 1055: ib 9388: [6, 27, 5, 8] # 1056: ib 9398: [7, 35, 5, 9] # 1057: ib 9403: [8, 43, 6, 9] # 1058: ib 9404: [7, 33, 6, 8] # 1059: ib 9420: [6, 27, 5, 8] # 1060: ib 9422: [7, 39, 6, 10] # 1061: ib 9430: [7, 35, 5, 8] # 1062: ib 9436: [7, 33, 6, 8] # 1063: ib 9438: [8, 41, 6, 10] # 1064: ib 9447: [8, 53, 7, 10] # 1065: ib 9452: [7, 39, 6, 10] # 1066: ib 9454: [8, 53, 7, 12] # 1067: ib 9455: [9, 63, 8, 12] # 1068: ib 9462: [8, 49, 6, 10] # 1069: ib 9463: [9, 59, 7, 10] # 1070: ib 9470: [9, 57, 7, 12] # 1071: ib 9471: [10, 67, 8, 12] # 1072: ib 9495: [7, 37, 6, 8] # 1073: ib 9496: [5, 17, 4, 6] # 1074: ib 9498: [6, 23, 5, 7] # 1075: ib 9499: [7, 29, 4, 10] # 1076: ib 9500: [6, 27, 5, 7] # 1077: ib 9519: [8, 53, 7, 9] # 1078: ib 9523: [7, 37, 6, 8] # 1079: ib 9563: [8, 43, 6, 10] # 1080: ib 9595: [9, 59, 7, 12] # 1081: ib 9624: [6, 27, 5, 8] # 1082: ib 9627: [8, 43, 5, 12] # 1083: ib 9655: [9, 59, 7, 10] # 1084: ib 9658: [8, 49, 7, 9] # 1085: ib 9659: [9, 59, 7, 12] # 1086: ib 9663: [10, 67, 8, 12] # 1087: ib 9690: [8, 49, 7, 10] # 1088: ib 9691: [9, 59, 7, 12] # 1089: ib 9694: [9, 57, 8, 10] # 1090: ib 9723: [10, 77, 9, 12] # 1091: ib 9724: [9, 63, 8, 10] # 1092: ib 9727: [11, 87, 10, 12] # 1093: ib 9746: [5, 19, 4, 7] # 1094: ib 9750: [6, 29, 5, 9] # 1095: ib 9753: [6, 27, 5, 7] # 1096: ib 9754: [6, 23, 5, 7] # 1097: ib 9775: [8, 53, 7, 9] # 1098: ib 9778: [6, 29, 5, 9] # 1099: ib 9782: [7, 41, 6, 11] # 1100: ib 9784: [6, 27, 5, 7] # 1101: ib 9786: [7, 35, 6, 9] # 1102: ib 9787: [8, 43, 6, 9] # 1103: ib 9788: [7, 39, 6, 8] # 1104: ib 9810: [6, 29, 5, 8] # 1105: ib 9811: [7, 37, 6, 8] # 1106: ib 9814: [7, 41, 6, 10] # 1107: ib 9816: [6, 27, 5, 7] # 1108: ib 9820: [7, 39, 6, 8] # 1109: ib 9822: [8, 49, 7, 10] # 1110: ib 9823: [9, 59, 7, 10] # 1111: ib 9842: [7, 41, 6, 10] # 1112: ib 9846: [8, 55, 7, 12] # 1113: ib 9848: [7, 39, 6, 9] # 1114: ib 9852: [8, 53, 7, 10] # 1115: ib 9854: [9, 65, 8, 12] # 1116: ib 9855: [10, 77, 8, 12] # 1117: ib 9876: [6, 27, 5, 7] # 1118: ib 9878: [7, 35, 5, 9] # 1119: ib 9879: [8, 43, 6, 9] # 1120: ib 9880: [6, 27, 5, 7] # 1121: ib 9881: [7, 39, 6, 9] # 1122: ib 9883: [8, 43, 6, 9] # 1123: ib 9884: [7, 33, 6, 8] # 1124: ib 9908: [7, 39, 6, 8] # 1125: ib 9909: [8, 53, 7, 9] # 1126: ib 9910: [8, 49, 6, 11] # 1127: ib 9940: [7, 39, 6, 8] # 1128: ib 9941: [8, 53, 7, 9] # 1129: ib 9942: [8, 49, 6, 10] # 1130: ib 9950: [9, 57, 8, 10] # 1131: ib 9951: [10, 67, 8, 12] # 1132: ib 9971: [9, 67, 8, 10] # 1133: ib 9972: [8, 53, 7, 9] # 1134: ib 9974: [9, 65, 7, 12] # 1135: ib 9982: [10, 75, 9, 12] # 1136: ib 9983: [11, 87, 10, 12] # 1137: ib 10003: [7, 37, 6, 8] # 1138: ib 10008: [6, 27, 5, 7] # 1139: ib 10012: [7, 39, 6, 8] # 1140: ib 10013: [8, 53, 7, 10] # 1141: ib 10015: [9, 59, 7, 10] # 1142: ib 10027: [8, 53, 7, 9] # 1143: ib 10111: [11, 97, 10, 12] # 1144: ib 10123: [8, 53, 7, 9] # 1145: ib 10132: [7, 39, 6, 8] # 1146: ib 10134: [8, 49, 7, 9] # 1147: ib 10136: [7, 39, 6, 9] # 1148: ib 10140: [8, 47, 7, 10] # 1149: ib 10141: [9, 63, 8, 12] # 1150: ib 10143: [10, 67, 8, 12] # 1151: ib 10163: [9, 67, 8, 10] # 1152: ib 10164: [8, 53, 7, 9] # 1153: ib 10167: [10, 77, 8, 12] # 1154: ib 10175: [11, 87, 10, 12] # 1155: ib 10196: [8, 53, 7, 9] # 1156: ib 10198: [9, 65, 8, 10] # 1157: ib 10207: [11, 87, 10, 12] # 1158: ib 10228: [9, 69, 8, 10] # 1159: ib 10231: [11, 97, 10, 12] # 1160: ib 10258: [4, 11, 3, 5] # 1161: ib 10262: [5, 19, 4, 7] # 1162: ib 10290: [5, 19, 4, 7] # 1163: ib 10294: [6, 29, 5, 9] # 1164: ib 10322: [5, 19, 4, 6] # 1165: ib 10326: [6, 29, 5, 8] # 1166: ib 10327: [7, 37, 6, 8] # 1167: ib 10354: [6, 29, 5, 8] # 1168: ib 10355: [7, 37, 6, 8] # 1169: ib 10358: [7, 41, 6, 10] # 1170: ib 10367: [9, 67, 8, 10] # 1171: ib 10375: [6, 27, 5, 7] # 1172: ib 10383: [7, 39, 6, 9] # 1173: ib 10390: [6, 23, 4, 7] # 1174: ib 10396: [6, 27, 5, 7] # 1175: ib 10398: [7, 35, 5, 9] # 1176: ib 10399: [8, 43, 6, 9] # 1177: ib 10422: [7, 35, 5, 9] # 1178: ib 10428: [7, 39, 6, 8] # 1179: ib 10429: [8, 53, 7, 9] # 1180: ib 10430: [8, 49, 6, 11] # 1181: ib 10454: [7, 35, 5, 8] # 1182: ib 10460: [7, 39, 6, 8] # 1183: ib 10461: [8, 53, 7, 9] # 1184: ib 10462: [8, 49, 6, 10] # 1185: ib 10471: [8, 53, 7, 10] # 1186: ib 10479: [9, 69, 8, 12] # 1187: ib 10486: [8, 49, 6, 10] # 1188: ib 10487: [9, 59, 7, 10] # 1189: ib 10492: [8, 53, 7, 9] # 1190: ib 10494: [9, 65, 7, 12] # 1191: ib 10495: [10, 77, 8, 12] # 1192: ib 10519: [7, 37, 6, 8] # 1193: ib 10547: [7, 37, 6, 8] # 1194: ib 10559: [9, 67, 8, 10] # 1195: ib 10652: [7, 39, 6, 8] # 1196: ib 10654: [8, 49, 7, 9] # 1197: ib 10679: [9, 59, 7, 10] # 1198: ib 10684: [8, 53, 7, 9] # 1199: ib 10687: [10, 77, 8, 12] # 1200: ib 10716: [8, 53, 7, 9] # 1201: ib 10718: [9, 65, 8, 10] # 1202: ib 10748: [9, 69, 8, 10] # 1203: ib 10751: [11, 97, 10, 12] # 1204: ib 10770: [5, 19, 4, 7] # 1205: ib 10774: [6, 29, 5, 9] # 1206: ib 10802: [6, 29, 5, 9] # 1207: ib 10806: [7, 41, 6, 11] # 1208: ib 10834: [6, 29, 5, 8] # 1209: ib 10835: [7, 37, 6, 8] # 1210: ib 10838: [7, 41, 6, 10] # 1211: ib 10847: [9, 67, 8, 10] # 1212: ib 10866: [7, 41, 6, 10] # 1213: ib 10870: [8, 55, 7, 12] # 1214: ib 10875: [9, 67, 8, 10] # 1215: ib 10900: [6, 27, 5, 7] # 1216: ib 10902: [7, 35, 5, 9] # 1217: ib 10903: [8, 43, 6, 9] # 1218: ib 10910: [8, 49, 7, 9] # 1219: ib 10932: [7, 39, 6, 8] # 1220: ib 10933: [8, 53, 7, 9] # 1221: ib 10934: [8, 49, 6, 11] # 1222: ib 10942: [9, 65, 8, 11] # 1223: ib 10964: [7, 39, 6, 8] # 1224: ib 10965: [8, 53, 7, 9] # 1225: ib 10966: [8, 49, 6, 10] # 1226: ib 10974: [9, 65, 8, 10] # 1227: ib 10995: [9, 67, 8, 10] # 1228: ib 10996: [8, 53, 7, 9] # 1229: ib 10998: [9, 65, 7, 12] # 1230: ib 11006: [10, 83, 9, 12] # 1231: ib 11007: [11, 97, 10, 12] # 1232: ib 11027: [7, 37, 6, 8] # 1233: ib 11039: [9, 67, 8, 10] # 1234: ib 11067: [9, 67, 8, 10] # 1235: ib 11156: [7, 39, 6, 8] # 1236: ib 11158: [8, 49, 7, 9] # 1237: ib 11187: [9, 67, 8, 10] # 1238: ib 11188: [8, 53, 7, 9] # 1239: ib 11191: [10, 77, 8, 12] # 1240: ib 11199: [11, 97, 10, 12] # 1241: ib 11220: [8, 53, 7, 9] # 1242: ib 11222: [9, 65, 8, 10] # 1243: ib 11252: [9, 69, 8, 10] # 1244: ib 11255: [11, 97, 10, 12] # 1245: ib 11282: [5, 19, 4, 7] # 1246: ib 11286: [6, 29, 5, 9] # 1247: ib 11288: [5, 17, 4, 6] # 1248: ib 11290: [6, 23, 4, 9] # 1249: ib 11291: [7, 29, 4, 9] # 1250: ib 11292: [6, 27, 5, 7] # 1251: ib 11294: [7, 35, 6, 9] # 1252: ib 11295: [8, 43, 6, 9] # 1253: ib 11314: [6, 29, 5, 9] # 1254: ib 11318: [7, 41, 6, 11] # 1255: ib 11320: [6, 27, 5, 8] # 1256: ib 11322: [7, 35, 5, 11] # 1257: ib 11323: [8, 43, 5, 11] # 1258: ib 11324: [7, 39, 6, 9] # 1259: ib 11326: [8, 49, 7, 11] # 1260: ib 11327: [9, 59, 7, 11] # 1261: ib 11346: [6, 29, 5, 8] # 1262: ib 11347: [7, 37, 6, 8] # 1263: ib 11350: [7, 41, 6, 10] # 1264: ib 11352: [6, 27, 5, 8] # 1265: ib 11354: [7, 35, 5, 12] # 1266: ib 11356: [7, 39, 6, 9] # 1267: ib 11378: [7, 41, 6, 10] # 1268: ib 11382: [8, 55, 7, 12] # 1269: ib 11384: [7, 39, 6, 10] # 1270: ib 11386: [8, 49, 6, 14] # 1271: ib 11388: [8, 53, 7, 11] # 1272: ib 11404: [6, 27, 5, 8] # 1273: ib 11414: [7, 35, 5, 9] # 1274: ib 11419: [8, 43, 6, 9] # 1275: ib 11420: [7, 33, 6, 8] # 1276: ib 11436: [7, 39, 6, 10] # 1277: ib 11446: [8, 49, 6, 11] # 1278: ib 11463: [8, 53, 7, 10] # 1279: ib 11468: [7, 39, 6, 10] # 1280: ib 11470: [8, 53, 7, 12] # 1281: ib 11471: [9, 63, 8, 12] # 1282: ib 11478: [8, 49, 6, 10] # 1283: ib 11479: [9, 59, 7, 10] # 1284: ib 11486: [9, 57, 7, 12] # 1285: ib 11487: [10, 67, 8, 12] # 1286: ib 11500: [8, 53, 7, 12] # 1287: ib 11502: [9, 69, 8, 14] # 1288: ib 11507: [9, 67, 8, 10] # 1289: ib 11510: [9, 65, 7, 12] # 1290: ib 11518: [10, 75, 8, 14] # 1291: ib 11535: [8, 53, 7, 9] # 1292: ib 11539: [7, 37, 6, 8] # 1293: ib 11567: [9, 69, 8, 11] # 1294: ib 11611: [9, 59, 7, 12] # 1295: ib 11643: [10, 77, 8, 14] # 1296: ib 11671: [9, 59, 7, 10] # 1297: ib 11674: [8, 49, 7, 9] # 1298: ib 11675: [9, 59, 7, 12] # 1299: ib 11679: [10, 67, 8, 12] # 1300: ib 11699: [9, 67, 8, 10] # 1301: ib 11708: [9, 63, 8, 10] # 1302: ib 11739: [10, 77, 9, 12] # 1303: ib 11740: [9, 63, 8, 10] # 1304: ib 11743: [11, 87, 10, 12] # 1305: ib 11773: [11, 101, 10, 12] # 1306: ib 11791: [8, 53, 7, 9] # 1307: ib 11794: [6, 29, 5, 9] # 1308: ib 11798: [7, 41, 6, 11] # 1309: ib 11800: [6, 27, 5, 7] # 1310: ib 11802: [7, 35, 6, 9] # 1311: ib 11803: [8, 43, 6, 9] # 1312: ib 11804: [7, 39, 6, 8] # 1313: ib 11823: [9, 69, 8, 11] # 1314: ib 11826: [7, 41, 6, 11] # 1315: ib 11830: [8, 55, 7, 13] # 1316: ib 11832: [7, 39, 6, 9] # 1317: ib 11834: [8, 49, 7, 11] # 1318: ib 11835: [9, 59, 7, 11] # 1319: ib 11836: [8, 53, 7, 10] # 1320: ib 11858: [7, 41, 6, 10] # 1321: ib 11862: [8, 55, 7, 12] # 1322: ib 11864: [7, 39, 6, 9] # 1323: ib 11868: [8, 53, 7, 10] # 1324: ib 11870: [9, 65, 8, 12] # 1325: ib 11871: [10, 77, 8, 12] # 1326: ib 11890: [8, 55, 7, 12] # 1327: ib 11894: [9, 71, 8, 14] # 1328: ib 11896: [8, 53, 7, 11] # 1329: ib 11900: [9, 69, 8, 12] # 1330: ib 11902: [10, 83, 9, 14] # 1331: ib 11903: [11, 97, 9, 14] # 1332: ib 11924: [7, 39, 6, 8] # 1333: ib 11925: [8, 53, 7, 9] # 1334: ib 11926: [8, 49, 6, 11] # 1335: ib 11956: [8, 53, 7, 9] # 1336: ib 11958: [9, 65, 7, 13] # 1337: ib 11987: [9, 67, 8, 10] # 1338: ib 11988: [8, 53, 7, 9] # 1339: ib 11990: [9, 65, 7, 12] # 1340: ib 11998: [10, 75, 9, 12] # 1341: ib 11999: [11, 87, 10, 12] # 1342: ib 12020: [9, 69, 8, 10] # 1343: ib 12022: [10, 83, 8, 14] # 1344: ib 12030: [11, 95, 10, 14] # 1345: ib 12043: [8, 53, 7, 9] # 1346: ib 12075: [9, 69, 8, 11] # 1347: ib 12127: [11, 97, 10, 12] # 1348: ib 12159: [12, 119, 11, 14] # 1349: ib 12179: [9, 67, 8, 10] # 1350: ib 12180: [8, 53, 7, 9] # 1351: ib 12183: [10, 77, 8, 12] # 1352: ib 12191: [11, 87, 10, 12] # 1353: ib 12212: [9, 69, 8, 10] # 1354: ib 12244: [9, 69, 8, 10] # 1355: ib 12247: [11, 97, 10, 12] # 1356: ib 12276: [10, 87, 9, 11] # 1357: ib 12277: [11, 107, 10, 12] # 1358: ib 12307: [5, 15, 3, 8] # 1359: ib 12314: [5, 19, 4, 6] # 1360: ib 12315: [6, 25, 4, 10] # 1361: ib 12317: [6, 29, 5, 7] # 1362: ib 12325: [5, 19, 4, 6] # 1363: ib 12327: [6, 29, 5, 8] # 1364: ib 12333: [6, 29, 5, 7] # 1365: ib 12338: [5, 19, 4, 6] # 1366: ib 12342: [6, 29, 5, 8] # 1367: ib 12343: [7, 35, 6, 8] # 1368: ib 12371: [6, 25, 4, 11] # 1369: ib 12378: [6, 29, 5, 7] # 1370: ib 12379: [7, 37, 5, 13] # 1371: ib 12389: [6, 29, 5, 7] # 1372: ib 12397: [7, 41, 6, 8] # 1373: ib 12402: [6, 29, 5, 7] # 1374: ib 12431: [7, 35, 5, 12] # 1375: ib 12432: [4, 11, 3, 5] # 1376: ib 12436: [5, 15, 3, 6] # 1377: ib 12437: [6, 23, 4, 7] # 1378: ib 12438: [6, 19, 3, 8] # 1379: ib 12440: [5, 19, 4, 7] # 1380: ib 12443: [7, 35, 5, 12] # 1381: ib 12444: [6, 25, 4, 9] # 1382: ib 12446: [7, 31, 4, 12] # 1383: ib 12447: [8, 39, 5, 12] # 1384: ib 12453: [6, 29, 5, 7] # 1385: ib 12464: [5, 19, 4, 6] # 1386: ib 12468: [6, 25, 4, 7] # 1387: ib 12470: [7, 31, 5, 8] # 1388: ib 12472: [6, 29, 5, 8] # 1389: ib 12476: [7, 37, 5, 10] # 1390: ib 12495: [8, 49, 6, 15] # 1391: ib 12496: [5, 19, 4, 7] # 1392: ib 12500: [6, 25, 4, 8] # 1393: ib 12502: [7, 31, 4, 11] # 1394: ib 12504: [6, 29, 5, 9] # 1395: ib 12507: [8, 49, 6, 15] # 1396: ib 12508: [7, 37, 5, 11] # 1397: ib 12510: [8, 45, 5, 15] # 1398: ib 12511: [9, 55, 6, 15] # 1399: ib 12528: [6, 29, 5, 8] # 1400: ib 12532: [7, 37, 5, 9] # 1401: ib 12536: [7, 41, 6, 10] # 1402: ib 12540: [8, 51, 6, 12] # 1403: ib 12549: [5, 19, 4, 6] # 1404: ib 12557: [6, 29, 5, 7] # 1405: ib 12562: [5, 19, 4, 6] # 1406: ib 12563: [6, 25, 5, 8] # 1407: ib 12565: [6, 29, 5, 8] # 1408: ib 12570: [6, 29, 5, 8] # 1409: ib 12571: [7, 37, 6, 10] # 1410: ib 12613: [6, 29, 5, 7] # 1411: ib 12621: [7, 41, 6, 8] # 1412: ib 12626: [6, 29, 5, 7] # 1413: ib 12634: [7, 41, 6, 9] # 1414: ib 12637: [8, 55, 7, 10] # 1415: ib 12677: [6, 29, 5, 7] # 1416: ib 12690: [6, 27, 5, 8] # 1417: ib 12694: [7, 31, 5, 8] # 1418: ib 12699: [8, 49, 7, 12] # 1419: ib 12703: [9, 55, 7, 12] # 1420: ib 12757: [8, 49, 6, 10] # 1421: ib 12805: [5, 19, 4, 6] # 1422: ib 12813: [6, 29, 5, 7] # 1423: ib 12818: [5, 19, 4, 6] # 1424: ib 12850: [6, 29, 5, 8] # 1425: ib 12851: [7, 37, 6, 8] # 1426: ib 12869: [6, 29, 5, 7] # 1427: ib 12877: [7, 41, 6, 8] # 1428: ib 12882: [6, 29, 5, 7] # 1429: ib 12914: [7, 41, 6, 9] # 1430: ib 12933: [6, 29, 5, 7] # 1431: ib 12944: [5, 19, 4, 6] # 1432: ib 12948: [6, 25, 4, 9] # 1433: ib 12952: [6, 29, 5, 8] # 1434: ib 12956: [7, 37, 6, 9] # 1435: ib 12958: [8, 45, 6, 12] # 1436: ib 12959: [9, 55, 7, 12] # 1437: ib 12976: [6, 29, 5, 7] # 1438: ib 12978: [7, 39, 6, 8] # 1439: ib 12980: [7, 37, 5, 10] # 1440: ib 12984: [7, 41, 6, 9] # 1441: ib 12988: [8, 51, 7, 10] # 1442: ib 13008: [6, 29, 5, 8] # 1443: ib 13012: [7, 37, 5, 11] # 1444: ib 13016: [7, 41, 6, 10] # 1445: ib 13020: [8, 51, 7, 11] # 1446: ib 13022: [9, 61, 7, 15] # 1447: ib 13023: [10, 73, 8, 15] # 1448: ib 13040: [7, 41, 6, 9] # 1449: ib 13044: [8, 51, 6, 12] # 1450: ib 13048: [8, 55, 7, 11] # 1451: ib 13052: [9, 67, 8, 12] # 1452: ib 13061: [6, 29, 5, 8] # 1453: ib 13069: [7, 41, 6, 9] # 1454: ib 13125: [7, 41, 6, 9] # 1455: ib 13133: [8, 55, 7, 10] # 1456: ib 13190: [7, 39, 6, 8] # 1457: ib 13207: [9, 55, 7, 12] # 1458: ib 13213: [9, 65, 8, 12] # 1459: ib 13215: [10, 73, 9, 12] # 1460: ib 13253: [8, 55, 7, 10] # 1461: ib 13317: [5, 19, 4, 6] # 1462: ib 13319: [6, 29, 5, 8] # 1463: ib 13325: [6, 29, 5, 7] # 1464: ib 13330: [5, 19, 4, 6] # 1465: ib 13334: [6, 29, 5, 8] # 1466: ib 13335: [7, 35, 6, 8] # 1467: ib 13336: [5, 17, 4, 6] # 1468: ib 13338: [6, 23, 4, 10] # 1469: ib 13340: [6, 27, 5, 7] # 1470: ib 13349: [6, 29, 5, 8] # 1471: ib 13362: [6, 29, 5, 8] # 1472: ib 13363: [7, 37, 6, 8] # 1473: ib 13368: [6, 27, 5, 8] # 1474: ib 13370: [7, 35, 5, 12] # 1475: ib 13381: [6, 29, 5, 7] # 1476: ib 13394: [6, 29, 5, 7] # 1477: ib 13400: [6, 27, 5, 8] # 1478: ib 13402: [7, 35, 5, 13] # 1479: ib 13404: [7, 39, 6, 9] # 1480: ib 13413: [7, 41, 6, 9] # 1481: ib 13415: [8, 55, 7, 11] # 1482: ib 13426: [7, 41, 6, 9] # 1483: ib 13430: [8, 55, 7, 11] # 1484: ib 13431: [9, 65, 8, 11] # 1485: ib 13432: [7, 39, 6, 10] # 1486: ib 13434: [8, 49, 6, 15] # 1487: ib 13445: [6, 29, 5, 7] # 1488: ib 13456: [5, 19, 4, 6] # 1489: ib 13460: [6, 25, 4, 7] # 1490: ib 13462: [7, 31, 5, 8] # 1491: ib 13486: [8, 49, 7, 12] # 1492: ib 13487: [9, 59, 8, 12] # 1493: ib 13488: [6, 29, 5, 7] # 1494: ib 13490: [7, 39, 6, 8] # 1495: ib 13492: [7, 37, 5, 8] # 1496: ib 13502: [9, 53, 7, 12] # 1497: ib 13503: [10, 63, 8, 12] # 1498: ib 13520: [6, 29, 5, 8] # 1499: ib 13524: [7, 37, 5, 9] # 1500: ib 13541: [8, 55, 7, 10] # 1501: ib 13552: [7, 41, 6, 9] # 1502: ib 13556: [8, 51, 6, 10] # 1503: ib 13558: [9, 61, 7, 11] # 1504: ib 13595: [8, 43, 6, 10] # 1505: ib 13627: [9, 59, 7, 12] # 1506: ib 13723: [9, 57, 7, 12] # 1507: ib 13755: [10, 75, 9, 12] # 1508: ib 13759: [11, 83, 10, 12] # 1509: ib 13842: [6, 29, 5, 8] # 1510: ib 13843: [7, 37, 6, 8] # 1511: ib 13848: [6, 27, 5, 7] # 1512: ib 13874: [7, 41, 6, 10] # 1513: ib 13878: [8, 55, 7, 12] # 1514: ib 13880: [7, 39, 6, 9] # 1515: ib 13886: [9, 65, 8, 12] # 1516: ib 13887: [10, 75, 8, 12] # 1517: ib 13906: [7, 41, 6, 9] # 1518: ib 13912: [7, 39, 6, 9] # 1519: ib 13938: [8, 55, 7, 11] # 1520: ib 13939: [9, 67, 8, 11] # 1521: ib 13944: [8, 53, 7, 11] # 1522: ib 13968: [6, 29, 5, 7] # 1523: ib 13970: [7, 39, 6, 8] # 1524: ib 13972: [7, 37, 5, 10] # 1525: ib 13983: [10, 63, 8, 12] # 1526: ib 14000: [7, 41, 6, 8] # 1527: ib 14004: [8, 51, 6, 11] # 1528: ib 14006: [9, 61, 7, 12] # 1529: ib 14014: [10, 71, 9, 12] # 1530: ib 14015: [11, 83, 10, 12] # 1531: ib 14032: [7, 41, 6, 9] # 1532: ib 14036: [8, 51, 6, 12] # 1533: ib 14064: [8, 55, 7, 10] # 1534: ib 14066: [9, 69, 8, 11] # 1535: ib 14068: [9, 67, 7, 13] # 1536: ib 14143: [11, 95, 10, 12] # 1537: ib 14239: [11, 83, 10, 12] # 1538: ib 14263: [11, 93, 10, 12] # 1539: ib 14341: [5, 19, 4, 6] # 1540: ib 14343: [6, 29, 5, 8] # 1541: ib 14349: [6, 29, 5, 7] # 1542: ib 14354: [5, 19, 4, 6] # 1543: ib 14358: [6, 29, 5, 8] # 1544: ib 14359: [7, 35, 6, 8] # 1545: ib 14373: [6, 29, 5, 8] # 1546: ib 14381: [7, 41, 6, 9] # 1547: ib 14386: [6, 29, 5, 8] # 1548: ib 14387: [7, 37, 6, 8] # 1549: ib 14405: [6, 29, 5, 7] # 1550: ib 14413: [7, 41, 6, 8] # 1551: ib 14418: [6, 29, 5, 7] # 1552: ib 14437: [7, 41, 6, 9] # 1553: ib 14439: [8, 55, 7, 11] # 1554: ib 14445: [8, 55, 7, 10] # 1555: ib 14450: [7, 41, 6, 9] # 1556: ib 14454: [8, 55, 7, 11] # 1557: ib 14455: [9, 65, 8, 11] # 1558: ib 14469: [6, 29, 5, 7] # 1559: ib 14480: [5, 19, 4, 6] # 1560: ib 14484: [6, 25, 4, 7] # 1561: ib 14486: [7, 31, 5, 8] # 1562: ib 14488: [6, 29, 5, 8] # 1563: ib 14492: [7, 37, 5, 10] # 1564: ib 14511: [9, 65, 8, 12] # 1565: ib 14512: [6, 29, 5, 7] # 1566: ib 14514: [7, 39, 6, 8] # 1567: ib 14516: [7, 37, 5, 8] # 1568: ib 14520: [7, 41, 6, 9] # 1569: ib 14524: [8, 51, 6, 11] # 1570: ib 14526: [9, 61, 7, 12] # 1571: ib 14527: [10, 73, 8, 12] # 1572: ib 14544: [6, 29, 5, 8] # 1573: ib 14548: [7, 37, 5, 9] # 1574: ib 14552: [7, 41, 6, 10] # 1575: ib 14556: [8, 51, 6, 12] # 1576: ib 14565: [8, 55, 7, 10] # 1577: ib 14576: [7, 41, 6, 9] # 1578: ib 14580: [8, 51, 6, 10] # 1579: ib 14582: [9, 61, 7, 11] # 1580: ib 14584: [8, 55, 7, 11] # 1581: ib 14588: [9, 67, 7, 13] # 1582: ib 14783: [11, 93, 10, 12] # 1583: ib 14866: [6, 29, 5, 8] # 1584: ib 14867: [7, 37, 6, 8] # 1585: ib 14898: [7, 41, 6, 10] # 1586: ib 14902: [8, 55, 7, 12] # 1587: ib 14907: [9, 67, 8, 10] # 1588: ib 14930: [7, 41, 6, 9] # 1589: ib 14962: [8, 55, 7, 11] # 1590: ib 14963: [9, 67, 8, 11] # 1591: ib 14992: [6, 29, 5, 7] # 1592: ib 14994: [7, 39, 6, 8] # 1593: ib 14996: [7, 37, 5, 10] # 1594: ib 15000: [7, 41, 6, 9] # 1595: ib 15004: [8, 51, 7, 10] # 1596: ib 15024: [7, 41, 6, 8] # 1597: ib 15028: [8, 51, 6, 11] # 1598: ib 15030: [9, 61, 7, 12] # 1599: ib 15032: [8, 55, 7, 10] # 1600: ib 15036: [9, 67, 8, 11] # 1601: ib 15038: [10, 79, 9, 12] # 1602: ib 15039: [11, 93, 10, 12] # 1603: ib 15056: [7, 41, 6, 9] # 1604: ib 15060: [8, 51, 6, 12] # 1605: ib 15064: [8, 55, 7, 11] # 1606: ib 15068: [9, 67, 8, 12] # 1607: ib 15088: [8, 55, 7, 10] # 1608: ib 15090: [9, 69, 8, 11] # 1609: ib 15092: [9, 67, 7, 13] # 1610: ib 15096: [9, 71, 8, 12] # 1611: ib 15100: [10, 85, 9, 13] # 1612: ib 15287: [11, 93, 10, 12] # 1613: ib 15365: [6, 29, 5, 8] # 1614: ib 15378: [6, 29, 5, 8] # 1615: ib 15379: [7, 37, 6, 8] # 1616: ib 15384: [6, 27, 5, 8] # 1617: ib 15386: [7, 35, 5, 12] # 1618: ib 15397: [7, 41, 6, 10] # 1619: ib 15410: [7, 41, 6, 10] # 1620: ib 15416: [7, 39, 6, 10] # 1621: ib 15418: [8, 49, 6, 14] # 1622: ib 15429: [7, 41, 6, 9] # 1623: ib 15431: [8, 55, 7, 11] # 1624: ib 15442: [7, 41, 6, 9] # 1625: ib 15446: [8, 55, 7, 11] # 1626: ib 15447: [9, 65, 8, 11] # 1627: ib 15448: [7, 39, 6, 10] # 1628: ib 15450: [8, 49, 6, 15] # 1629: ib 15461: [8, 55, 7, 11] # 1630: ib 15474: [8, 55, 7, 11] # 1631: ib 15475: [9, 67, 8, 11] # 1632: ib 15480: [8, 53, 7, 12] # 1633: ib 15482: [9, 65, 7, 17] # 1634: ib 15502: [8, 49, 7, 12] # 1635: ib 15503: [9, 59, 8, 12] # 1636: ib 15504: [6, 29, 5, 7] # 1637: ib 15506: [7, 39, 6, 8] # 1638: ib 15508: [7, 37, 5, 8] # 1639: ib 15518: [9, 53, 7, 12] # 1640: ib 15519: [10, 63, 8, 12] # 1641: ib 15532: [8, 53, 7, 12] # 1642: ib 15533: [9, 67, 8, 12] # 1643: ib 15536: [7, 41, 6, 8] # 1644: ib 15540: [8, 51, 6, 9] # 1645: ib 15544: [8, 51, 7, 10] # 1646: ib 15548: [9, 61, 7, 12] # 1647: ib 15557: [8, 55, 7, 10] # 1648: ib 15568: [7, 41, 6, 9] # 1649: ib 15572: [8, 51, 6, 10] # 1650: ib 15574: [9, 61, 7, 11] # 1651: ib 15596: [9, 69, 8, 14] # 1652: ib 15600: [8, 55, 7, 10] # 1653: ib 15602: [9, 69, 8, 11] # 1654: ib 15604: [9, 67, 7, 11] # 1655: ib 15608: [9, 67, 8, 12] # 1656: ib 15612: [10, 79, 8, 14] # 1657: ib 15643: [9, 59, 7, 12] # 1658: ib 15675: [10, 77, 8, 14] # 1659: ib 15771: [10, 75, 9, 12] # 1660: ib 15775: [11, 83, 10, 12] # 1661: ib 15805: [11, 97, 10, 12] # 1662: ib 15890: [7, 41, 6, 10] # 1663: ib 15894: [8, 55, 7, 12] # 1664: ib 15896: [7, 39, 6, 9] # 1665: ib 15902: [9, 65, 8, 12] # 1666: ib 15903: [10, 75, 8, 12] # 1667: ib 15922: [8, 55, 7, 12] # 1668: ib 15928: [8, 53, 7, 11] # 1669: ib 15932: [9, 69, 8, 12] # 1670: ib 15954: [8, 55, 7, 11] # 1671: ib 15955: [9, 67, 8, 11] # 1672: ib 15960: [8, 53, 7, 11] # 1673: ib 15986: [9, 71, 8, 13] # 1674: ib 15990: [10, 89, 9, 15] # 1675: ib 15992: [9, 69, 8, 13] # 1676: ib 15996: [10, 87, 9, 14] # 1677: ib 16016: [7, 41, 6, 8] # 1678: ib 16020: [8, 51, 6, 11] # 1679: ib 16022: [9, 61, 7, 12] # 1680: ib 16030: [10, 71, 9, 12] # 1681: ib 16031: [11, 83, 10, 12] # 1682: ib 16048: [8, 55, 7, 9] # 1683: ib 16052: [9, 67, 7, 12] # 1684: ib 16056: [9, 67, 8, 11] # 1685: ib 16060: [10, 79, 9, 12] # 1686: ib 16080: [8, 55, 7, 10] # 1687: ib 16082: [9, 69, 8, 11] # 1688: ib 16084: [9, 67, 7, 13] # 1689: ib 16112: [9, 71, 8, 11] # 1690: ib 16116: [10, 85, 8, 14] # 1691: ib 16118: [11, 99, 9, 15] # 1692: ib 16120: [10, 85, 9, 13] # 1693: ib 16124: [11, 99, 10, 14] # 1694: ib 16159: [11, 95, 10, 12] # 1695: ib 16279: [11, 93, 10, 12] # 1696: ib 16309: [11, 103, 10, 12] # That's all, not computed further.