Cheers,
19 NEW PRENUMBERED SEQUENCES (A106442-A106447 A106451-A106457 A106490-A106495)
follow:
%I A106442
%S A106442 0,1,2,3,4,7,6,11,8,5,14,13,12,19,22,9,16,25,10,31,28,29,26,37,24,21,
%T A106442 38,15,44,41,18,47,128,23,50,49,20,55,62,53,56,59,58,61,52,27,74,67,
%U A106442 192,69,42,43,76,73,30,35,88,33,82,87,36,91,94,39,64,121,46,97,100,111,98
%N A106442 Exponent-recursed cross-domain bijection from N to GF(2)[X]. Position of A075166(n) in A106456.
%F A106442 a(0)=0, a(1)=1, a(p_i) = A014580(i) for primes p_i with index i, and for composites n = p_i^e_i * p_j^e_j * p_k^e_k * ..., a(n) = A048723(a(p_i),a(e_i)) X A048723(a(p_j),a(1+e_j)-1) X A048723(a(p_k),a(1+e_k)-1) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720), and A048723(n,y) raises the nth GF(2)[X] polynomial to the y:th power. Here p_i is the most significant prime in the factorization of n; its exponent e_i is not incremented before the recursion step, while the exponents of less significant primes e_j, e_k, ... are incremented by one before recursing, and the result of the recursion is decremented by one before use.
%e A106442 a(5) = 7, as 5 is the 3rd prime, and the third irreducible GF(2)[X] polynomial x^2+x+1 is encoded as A014580(3) = 7. a(32) = a(2^5) = A048723(A014580(1),a(5)) = A048723(2,7) = 128. a(48) = a(3 * 2^4) = 3 X A048723(2,a(4+1)-1) = 3 X A048723(2,7-1) = 3 X 64 = 192.
%C A106442 This map from the multiplicative domain of N to that of GF(2)[X] preserves Catalan-family structures, e.g. A106454(n) = a(A075164(n)), A075163(n) = A106453(a(n)), A075165(n) = A106455(a(n)), A075166(n) = A106456(a(n)), A075167(n) = A106457(a(n)). Shares with A091202 and A106444 the property that maps A000040(n) to A014580(n). Differs from the former for the first time at n=32, where A091202(32)=32, while a(32)=128. Differs from the latter for the first time at n=48, where A106444(48)=48, while a(48)=192.
%H A106442 A. Karttunen, Scheme-program for computing this sequence.
%H A106442 Index entries for sequences that are permutations of the natural numbers
%Y A106442 Inverse: A106443. a(n) = A106454(A075163(n)).
%K A106442 nonn
%O A106442 0,3
%A A106442 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106443
%S A106443 0,1,2,3,4,9,6,5,8,15,18,7,12,11,10,27,16,81,30,13,36,25,14,33,24,17,
%T A106443 22,45,20,21,54,19,512,57,162,55,60,23,26,63,72,29,50,51,28,135,66,31,
%U A106443 768,35,34,19683,44,39,90,37,40,99,42,41,108,43,38,75,64,225,114,47
%N A106443 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Position of A106456(n) in A075166.
%F A106443 a(0)=0, a(1)=1. For irreducible GF(2)[X] polynomials ir_i with index i (i.e. A014580(i)), a(ir_i) = A000040(i), and for composite polynomials n = A048723(ir_i,e_i) X A048723(ir_j,e_j) X A048723(ir_k,e_k) X ..., a(n) = a(ir_i)^a(e_i) * a(ir_j)^(a(1+e_j)-1) * a(ir_k)^(a(1+e_k)-1) * ... = A000040(i)^a(e_i) * A000040(j)^(a(1+e_j)-1) * A000040(k)^(a(1+e_k)-1) , where X stands for carryless multiplication of GF(2)[X] polynomials (A048720), and A048723(n,y) raises the nth GF(2)[X] polynomial to the y:th power, while * is the ordinary multiplication, and ^ is the ordinary exponentiation. Here ir_i is the most significant (largest) irreducible polynomial in the factorization of n; its exponent e_i is not incremented before the recursion step, while the exponents of less significant factors e_j, e_k, ... are incremented by one before recursing, and the result of the recursion is decremented by one before use.
%e A106443 a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3), and the square of the second prime is 3^2=9. a(32) = a(A048723(2,5)) = 2^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = 3 * 2^(a(4+1)-1) = 3 * 2^(9-1) = 3 * 256 = 768.
%C A106443 This map from the multiplicative domain of GF(2)[X] to that of N preserves Catalan-family structures, e.g. A075164(n) = a(A106454(n)), A106453(n) = A075163(a(n)), A106455(n) = A075165(a(n)), A106456(n) = A075166(a(n)), A106457(n) = A075167(a(n)). Shares with A091203 and A106445 the property that maps A014580(n) to A000040(n). Differs from the former for the first time at n=32, where A091203(32)=32, while a(32)=512. Differs from the latter for the first time at n=48, where A106445(48)=48, while a(48)=768.
%H A106443 A. Karttunen, Scheme-program for computing this sequence.
%H A106443 Index entries for sequences that are permutations of the natural numbers
%Y A106443 Inverse: A106442. a(n) = A075164(A106453(n)).
%K A106443 nonn
%O A106443 0,3
%A A106443 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106444
%S A106444 0,1,2,3,4,7,6,11,8,5,14,13,12,19,22,9,16,25,10,31,28,29,26,37,24,21,
%T A106444 38,15,44,41,18,47,128,23,50,49,20,55,62,53,56,59,58,61,52,27,74,67,
%U A106444 48,69,42,43,76,73,30,35,88,33,82,87,36,91,94,39,64,121,46,97,100,111
%N A106444 Exponent-recursed cross-domain bijection from N to GF(2)[X]. Variant of A091202 and A106442.
%F A106444 a(0)=0, a(1)=1, a(p_i) = A014580(i) for primes p_i with index i, and for composites n = p_i^e_i * p_j^e_j * p_k^e_k * ..., a(n) = A048723(a(p_i),a(e_i)) X A048723(a(p_j),a(e_j)) X A048723(a(p_k),a(e_k)) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720), and A048723(n,y) raises the nth GF(2)[X] polynomial to the y:th power.
%e A106444 a(5) = 7, as 5 is the 3rd prime, and the third irreducible GF(2)[X] polynomial x^2+x+1 is encoded as A014580(3) = 7. a(32) = a(2^5) = A048723(A014580(1),a(5)) = A048723(2,7) = 128. a(48) = a(3 * 2^4) = 3 X A048723(2,a(4)) = 3 X A048723(2,4) = 3 X 16 = 48.
%C A106444 This map from the multiplicative domain of N to that of GF(2)[X] preserves 'superfactorized' structures, e.g. A106490(n) = A106493(a(n)), A106491(n) = A106494(a(n)), A064372(n) = A106495(a(n)). Shares with A091202 and A106442 the property that maps A000040(n) to A014580(n). Differs from A091202 for the first time at n=32, where A091202(32)=32, while a(32)=128. Differs from A106442 for the first time at n=48, where A106442(48)=192, while a(48)=48. Differs from A106446 for the first time at n=11, where A106446(11)=25, while a(11)=13.
%H A106444 A. Karttunen, Scheme-program for computing this sequence.
%H A106444 Index entries for sequences that are permutations of the natural numbers
%Y A106444 Inverse: A106445.
%K A106444 nonn
%O A106444 0,3
%A A106444 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106445
%S A106445 0,1,2,3,4,9,6,5,8,15,18,7,12,11,10,27,16,81,30,13,36,25,14,33,24,17,
%T A106445 22,45,20,21,54,19,512,57,162,55,60,23,26,63,72,29,50,51,28,135,66,31,
%U A106445 48,35,34,19683,44,39,90,37,40,99,42,41,108,43,38,75,64,225,114,47
%N A106445 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091203 and A106443.
%F A106445 a(0)=0, a(1)=1. For irreducible GF(2)[X] polynomials ir_i with index i (i.e. A014580(i)), a(ir_i) = A000040(i), and for composite polynomials n = A048723(ir_i,e_i) X A048723(ir_j,e_j) X A048723(ir_k,e_k) X ..., a(n) = a(ir_i)^a(e_i) * a(ir_j)^a(e_j) * a(ir_k)^a(e_k) * ... = A000040(i)^a(e_i) * A000040(j)^a(e_j) * A000040(k)^a(e_k) , where X stands for carryless multiplication of GF(2)[X] polynomials (A048720), and A048723(n,y) raises the nth GF(2)[X] polynomial to the y:th power, while * is the ordinary multiplication, and ^ is the ordinary exponentiation.
%e A106445 a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3), and the square of the second prime is 3^2=9. a(32) = a(A048723(2,5)) = 2^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = 3 * 2^a(4) = 3 * 2^4 = 3 * 16 = 48.
%C A106445 This map from the multiplicative domain of GF(2)[X] to that of N preserves 'superfactorized' structures, e.g. A106493(n) = A106490(a(n)), A106494(n) = A106491(a(n)), A106495(n) = A064372(a(n)). Shares with A091203 and A106443 the property that maps A014580(n) to A000040(n). Differs from the plain variant A091203 for the first time at n=32, where A091203(32)=32, while a(32)=512. Differs from the variant A106443 for the first time at n=48, where A106443(48)=768, while a(48)=48. Differs from a yet deeper variant A106447 for the first time at n=13, where A106447(13)=23, while a(13)=11.
%H A106445 A. Karttunen, Scheme-program for computing this sequence.
%H A106445 Index entries for sequences that are permutations of the natural numbers
%Y A106445 Inverse: A106444.
%K A106445 nonn
%O A106445 0,3
%A A106445 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106446
%S A106446 0,1,2,3,4,7,6,11,8,5,14,25,12,19,22,9,16,47,10,31,28,29,50,13,24,21,
%T A106446 38,15,44,61,18,137,128,43,94,49,20,55,62,53,56,97,58,115,100,27,26,
%U A106446 37,48,69,42,113,76,73,30,79,88,33,122,319,36,41,274,39,64,121,86,185
%N A106446 Doubly-recursed cross-domain bijection from N to GF(2)[X]. Variant of A091204 and A106444.
%F A106446 a(0)=0, a(1)=1, a(p_i) = A014580(a(i)) for primes p_i with index i, and for composites n = p_i^e_i * p_j^e_j * p_k^e_k * ..., a(n) = A048723(a(p_i),a(e_i)) X A048723(a(p_j),a(e_j)) X A048723(a(p_k),a(e_k)) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720), and A048723(n,y) raises the nth GF(2)[X] polynomial to the y:th power.
%e A106446 a(5) = 7, as 5 is the 3rd prime, a(3)=3, and the third irreducible GF(2)[X] polynomial x^2+x+1 is encoded as A014580(3) = 7. a(11) = 25, as 11 is the 5th prime, a(5)=7, and the seventh irreducible GF(2)[X] polynomial x^4+x^3+1 is encoded as A014580(7) = 25. a(32) = a(2^5) = A048723(a(2),a(5)) = A048723(2,7) = 128.
%C A106446 Differs from A091204 for the first time at n=32, where A091204(32)=32, while a(32)=128. Differs from A106444 for the first time at n=11, where A106444(11)=13, while a(11)=25.
%H A106446 A. Karttunen, Scheme-program for computing this sequence.
%H A106446 Index entries for sequences that are permutations of the natural numbers
%Y A106446 Inverse: A106447. Variant: A091204.
%K A106446 nonn
%O A106446 0,3
%A A106446 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106447
%S A106447 0,1,2,3,4,9,6,5,8,15,18,7,12,23,10,27,16,81,30,13,36,25,14,69,24,11,
%T A106447 46,45,20,21,54,19,512,57,162,115,60,47,26,63,72,61,50,33,28,135,138,
%U A106447 17,48,35,22,19683,92,39,90,37,40,207,42,83,108,29,38,75,64,225,114
%N A106447 Doubly-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091205 and A106445.
%F A106447 a(0)=0, a(1)=1. For irreducible GF(2)[X] polynomials ir_i with index i (i.e. A014580(i)), a(ir_i) = A000040(a(i)), and for composite polynomials n = A048723(ir_i,e_i) X A048723(ir_j,e_j) X A048723(ir_k,e_k) X ..., a(n) = a(ir_i)^a(e_i) * a(ir_j)^a(e_j) * a(ir_k)^a(e_k) * ... = A000040(a(i))^a(e_i) * A000040(a(j))^a(e_j) * A000040(a(k))^a(e_k) , where X stands for carryless multiplication of GF(2)[X] polynomials (A048720), and A048723(n,y) raises the nth GF(2)[X] polynomial to the y:th power, while * is the ordinary multiplication, and ^ is the ordinary exponentiation.
%e A106447 a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3), a(2)=2, and the square of the second prime is 3^2=9. a(13) = a(A014580(5)) = A000040(a(5)) = A000040(9) = 23. a(32) = a(A048723(2,5)) = a(2)^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = a(3) * a(2)^a(4) = 3 * 2^4 = 3 * 16 = 48.
%C A106447 Differs from A091205 for the first time at n=32, where A091205(32)=32, while a(32)=512. Differs from A106445 for the first time at n=13, where A106445(13)=11, while a(13)=23.
%H A106447 A. Karttunen, Scheme-program for computing this sequence.
%H A106447 Index entries for sequences that are permutations of the natural numbers
%Y A106447 Inverse: A106446. Variant: A091205.
%K A106447 nonn
%O A106447 0,3
%A A106447 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106451
%S A106451 0,1,2,3,6,5,4,7,11,15,9,12,23,10,16,8,19,29,65,40,14,24,67,13,197,66,
%T A106451 30,26,25,43,626,20,2058,52,70,82,2056,198,72,41,6918,38,628,68,33,
%U A106451 203,23714,34,28,627,53,200,199,85,82500,27,204,71,290512,124,1033412
%N A106451 Position of A106455(n+1) in A014486.
%C A106451 See A106456.
%H A106451 A. Karttunen, Scheme-program for computing this sequence.
%H A106451 Index entries for sequences that are permutations of the natural numbers
%Y A106451 Inverse: A106452. a(n) = A106453(n+1)-1. GF(2)[X]-analogue of A075161.
%K A106451 nonn
%O A106451 0,3
%A A106451 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%S A106452 0,1,2,3,6,5,4,7,15,10,13,8,11,23,20,9,14,127,63,16,31,255,65535,12,
%T A106452 21,28,27,55,48,17,26,191,95,44,47,383,98303,68,41,62,19,39,106,29,
%U A106452 254,2047,16383,84,511,4095,16777215,272,33,50,2097151,1023,256,32767
%I A106452
%N A106452 Position of A014486(n) in A106455, minus one.
%C A106452 See A106456. The next term, a(58) = 340282366920938463463374607431768211455 = (2^128) - 1 as A063171(58) = 1110101000, 11010100 = A063171(17), a(17)=127 and 127+1 = 128.
%H A106452 A. Karttunen, Scheme-program for computing this sequence.
%H A106452 Index entries for sequences that are permutations of the natural numbers
%Y A106452 Inverse: A106451. a(n) = A106454(n+1)-1. GF(2)[X]-analogue of A075162.
%K A106452 nonn
%O A106452 0,3
%A A106452 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106453
%S A106453 1,2,3,4,7,6,5,8,12,16,10,13,24,11,17,9,20,30,66,41,15,25,68,14,198,
%T A106453 67,31,27,26,44,627,21,2059,53,71,83,2057,199,73,42,6919,39,629,69,34,
%U A106453 204,23715,35,29,628,54,201,200,86,82501,28,205,72,290513,125,1033413
%N A106453 Position of A106455(n) in A014486 plus one.
%C A106453 See A106456.
%H A106453 A. Karttunen, Scheme-program for computing this sequence.
%H A106453 Index entries for sequences that are permutations of the natural numbers
%Y A106453 Inverse: A106454. a(n) = A075163(A106443(n)). a(n) = A106451(n-1)+1. GF(2)[X]-analogue of A075163.
%K A106453 nonn
%O A106453 1,2
%A A106453 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106454
%S A106454 1,2,3,4,7,6,5,8,16,11,14,9,12,24,21,10,15,128,64,17,32,256,65536,13,
%T A106454 22,29,28,56,49,18,27,192,96,45,48,384,98304,69,42,63,20,40,107,30,
%U A106454 255,2048,16384,85,512,4096,16777216,273,34,51,2097152,1024,257,32768
%N A106454 Position of A014486(n-1) in A106455.
%C A106454 See A106456.
%H A106454 A. Karttunen, Scheme-program for computing this sequence.
%H A106454 Index entries for sequences that are permutations of the natural numbers
%Y A106454 Inverse: A106453. a(n) = A106442(A075164(n)). a(n) = A106452(n-1)+1. GF(2)[X]-analogue of A075164.
%K A106454 nonn
%O A106454 1,2
%A A106454 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106455
%S A106455 0,2,10,12,50,44,42,52,178,204,170,180,682,172,210,56,226,716,2730,
%T A106455 820,202,684,2738,184,10922,2732,722,692,690,844,43690,228,174770,908,
%U A106455 2762,2868,174762,10924,2770,824,699050,812,43698,2740,738,10956
%N A106455 Sequence A106456 interpreted as binary numbers and converted to decimal.
%H A106455 A. Karttunen, Scheme-program for computing this sequence.
%Y A106455 a(n) = A075165(A106443(n)). Permutation of A014486. Same sequence shown in binary: A106456. The binary width of each term / 2 is given by A106457. GF(2)[X]-analogue of A075165.
%K A106455 nonn
%O A106455 1,2
%A A106455 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106456
%S A106456 0,10,1010,1100,110010,101100,101010,110100,10110010,11001100,
%T A106456 10101010,10110100,1010101010,10101100,11010010,111000,11100010,
%U A106456 1011001100,101010101010,1100110100,11001010,1010101100,101010110010
%N A106456 Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the GF(2)[X] factorization of n.
%C A106456 Note that we recurse on the exponent + 1 for all other irreducible polynomials except the largest one in the GF(2)[X] factorization. Thus for 6 = A048723(3,1) X A048723(2,1) we construct a tree by joining trees 1 and 2 with a new root node, for 7 = A048723(7,1) X A048723(3,0) X A048723(2,0) we join three 1-trees (single leaves) with a new root node, for 8 = A048273(2,3) we add a single edge below tree 3, and for 9 = A048723(7,1) X A048723(3,1) X A048273(2,0) we connect the trees 1 and 2 and 1 with a new root node.
%e A106456 The rooted plane trees encoded here are:
%e A106456 .....................o....o..........o.........o...o....o.....
%e A106456 .....................|....|..........|..........\./.....|.....
%e A106456 .......o....o...o....o....o...o..o...o..o.o.o....o....o.o.o...
%e A106456 .......|.....\./.....|.....\./....\./....\|/.....|.....\|/....
%e A106456 *......*......*......*......*......*......*......*......*.....
%e A106456 1......2......3......4......5......6......7......8......9.....
%H A106456 A. Karttunen, Scheme-program for computing this sequence.
%Y A106456 a(n) = A007088(A106455(n)) = A075166(A106443(n)). GF(2)[X]-analogue of A075166. Permutation of A063171. Same sequence shown in decimal: A106455. The digital length of each term / 2 (the number of o-nodes in the corresponding trees) is given by A106457. C.f. A106451-A106454.
%K A106456 nonn
%O A106456 1,2
%A A106456 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106457
%S A106457 0,1,2,2,3,3,3,3,4,4,4,4,5,4,4,3,4,5,6,5,4,5,6,4,7,6,5,5,5,5,8,4,9,5,
%T A106457 6,6,9,7,6,5,10,5,8,6,5,7,11,5,5,8,5,7,7,6,12,5,7,6,13,6,14,9,5,4,6,
%U A106457 10,15,6,5,7,15,6,16,10,7,8,14,7,8,6,6,11,6,6,5,9,17,6,13,6,18,8,9,12
%N A106457 Number of edges in each rooted plane tree produced with the GF(2)[X] factorization unranking algorithm presented in A106456.
%C A106457 Also the digital length of A106456(n)/2. Each value v occurs A000108(v) times.
%H A106457 A. Karttunen, Scheme-program for computing this sequence.
%Y A106457 a(n) = A075167(A106443(n)). Permutation of A072643.
%K A106457 nonn
%O A106457 1,3
%A A106457 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106490
%S A106490 0,1,1,2,1,2,1,2,2,2,1,3,1,2,2,3,1,3,1,3,2,2,1,3,2,2,2,3,1,3,1,2,2,2,
%T A106490 2,4,1,2,2,3,1,3,1,3,3,2,1,4,2,3,2,3,1,3,2,3,2,2,1,4,1,2,3,3,2,3,1,3,
%U A106490 2,3,1,4,1,2,3,3,2,3,1,4,3,2,1,4,2,2,2,3,1,4,2,3,2,2,2,3,1,3,3,4,1,3
%N A106490 Total number of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.
%C A106490 Quetian Superfactorization proceeds by factoring a natural number to its unique prime-exponent factorization (p1^e1 * p2^e2 * ... pj^ej) and then factoring recursively each of the (non-zero) exponents in similar manner, until unity-exponents are finally encountered.
%e A106490 a(64) = 3, as 64 = 2^6 = 2^(2^1*3^1), and there are three non-1 nodes in that superfactorization. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 5. a(65536) = a(2^(2^(2^(2^1)))) = 4.
%H A106490 A. Karttunen, Scheme-program for computing this sequence.
%Y A106490 a(n) = A106493(A106444(n)). a(n) = A106491(n)-A064372(n). C.f. also A106492. After n=1 differs from A038548 for the first time at n=24, where A038548(24)=4, while a(24)=3.
%K A106490 nonn
%O A106490 1,4
%A A106490 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi) May 09 2005 based on Leroy Quet's (qq-quet(AT)mindspring.com) message (Subject: 'Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003.
%I A106491
%S A106491 1,2,2,3,2,4,2,3,3,4,2,5,2,4,4,4,2,5,2,5,4,4,2,5,3,4,3,5,2,6,2,3,4,4,
%T A106491 4,6,2,4,4,5,2,6,2,5,5,4,2,6,3,5,4,5,2,5,4,5,4,4,2,7,2,4,5,5,4,6,2,5,
%U A106491 4,6,2,6,2,4,5,5,4,6,2,6,4,4,2,7,4,4,4,5,2,7,4,5,4,4,4,5,2,5,5,6,2,6
%N A106491 Total number of bases and exponents in Quetian Superfactorization of n, including the unity-exponents at the tips of branches.
%e A106491 a(64) = 5, as 64 = 2^6 = 2^(2^1*3^1), and there are 5 nodes in that superfactorization. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 8. See comments at A106490.
%H A106491 A. Karttunen, Scheme-program for computing this sequence.
%Y A106491 a(n) = A106494(A106444(n)). a(n) = A106490(n)+A064372(n). C.f. also A106492.
%K A106491 nonn
%O A106491 1,2
%A A106491 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005 based on Leroy Quet's (qq-quet(AT)mindspring.com) message (Subject: 'Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003.
%I A106492
%S A106492 0,2,3,4,5,5,7,5,5,7,11,7,13,9,8,6,17,7,19,9,10,13,23,8,7,15,6,11,29,
%T A106492 10,31,7,14,19,12,9,37,21,16,10,41,12,43,15,10,25,47,9,9,9,20,17,53,8,
%U A106492 16,12,22,31,59,12,61,33,12,7,18,16,67,21,26,14,71,10,73,39,10,23,18
%N A106492 Total sum of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.
%e A106492 a(64) = 7, as 64 = 2^6 = 2^(2^1*3^1), and 2+2+3=7. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 2+3+3+2+5 = 15. See comments at A106490.
%H A106492 A. Karttunen, Scheme-program for computing this sequence.
%Y A106492 C.f. A106490-A106491.
%K A106492 nonn
%O A106492 1,2
%A A106492 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005 based on Leroy Quet's (qq-quet(AT)mindspring.com) message (Subject: 'Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003.
%I A106493
%S A106493 0,1,1,2,2,2,1,2,2,3,1,3,1,2,2,3,3,3,1,4,2,2,2,3,1,2,3,3,2,3,1,3,2,4,
%T A106493 2,4,1,2,3,4,1,3,2,3,3,3,1,4,2,2,3,3,2,4,1,3,3,3,1,4,1,2,3,3,4,3,1,5,
%U A106493 2,3,2,4,1,2,3,3,2,4,2,5,2,2,3,4,3,3,1,3,2,4,1,4,2,2,3,4,1,3,3,3,3,4
%N A106493 Total number of bases and exponents in GF(2)[X] Superfactorization of n, excluding the unity-exponents at the tips of branches.
%C A106493 GF(2)[X] Superfactorization proceeds in a manner analogous to normal superfactorization explained in A106490, but using factorization in domain GF(2)[X], instead of normal integer factorization in N.
%e A106493 a(64) = 3, as 64 = A048723(2,6) = A048723(2,(A048723(2,1) X A048723(3,1))), and there are three non-1 nodes in that superfactorization. Similarly, for 27 = 5x7 = A048723(3,2) X A048273(7,1) we get a(27) = 3. The operation X stands for GF(2)[X] multiplication defined in A048720, while A048723(n,y) raises the nth GF(2)[X] polynomial to the y:th power.
%H A106493 A. Karttunen, Scheme-program for computing this sequence.
%Y A106493 a(n) = A106490(A106445(n)). a(n) = A106494(n)-A106495(n).
%K A106493 nonn
%O A106493 1,4
%A A106493 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106494
%S A106494 1,2,2,3,3,4,2,3,4,5,2,5,2,4,3,4,4,6,2,6,3,4,4,5,2,4,5,5,4,5,2,4,4,6,
%T A106494 4,7,2,4,5,6,2,5,4,5,5,6,2,6,4,4,4,5,4,7,2,5,5,6,2,6,2,4,5,5,6,6,2,7,
%U A106494 3,6,4,7,2,4,5,5,4,7,4,7,3,4,6,6,5,6,2,5,4,7,2,7,4,4,5,6,2,6,5,5,6,6
%N A106494 Total number of bases and exponents in GF(2)[X] Superfactorization of n, including the unity-exponents at the tips of branches.
%C A106494 See comments at A106493.
%e A106494 a(64) = 5, as 64 = A048723(2,6) = A048723(2,(A048723(2,1) X A048723(3,1))), and there are five nodes in that superfactorization. Similarly, for 27 = 5x7 = A048723(3,A048723(2,1)) X A048273(7,1) we get a(27) = 5. The operation X stands for GF(2)[X] multiplication defined in A048720, while A048723(n,y) raises the nth GF(2)[X] polynomial to the y:th power.
%H A106494 A. Karttunen, Scheme-program for computing this sequence.
%Y A106494 a(n) = A106491(A106445(n)). a(n) = A106493(n)+A106495(n).
%K A106494 nonn
%O A106494 1,2
%A A106494 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
%I A106495
%S A106495 1,1,1,1,1,2,1,1,2,2,1,2,1,2,1,1,1,3,1,2,1,2,2,2,1,2,2,2,2,2,1,1,2,2,
%T A106495 2,3,1,2,2,2,1,2,2,2,2,3,1,2,2,2,1,2,2,3,1,2,2,3,1,2,1,2,2,2,2,3,1,2,
%U A106495 1,3,2,3,1,2,2,2,2,3,2,2,1,2,3,2,2,3,1,2,2,3,1,3,2,2,2,2,1,3,2,2,3,2
%N A106495 Number of leaves in GF(2)[X] superfactorization of n.
%F A106495 a(n) = A106494(n)-A106493(n).
%H A106495 A. Karttunen, Scheme-program for computing this sequence.
%Y A106495 a(n) = A064372(A106445(n)). After n=1 differs from A091221 for the first time at n=64, where A091221(64)=1, while a(64)=2.
%K A106495 nonn
%O A106495 1,6
%A A106495 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), May 09 2005
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Yours,
Antti Karttunen