Cheers,
38 NEW PRE-NUMBERED SEQUENCES A112045-A112082:
%I A112045
%S A112045 1,2,3,5,8,10,13,15,19,24,26,31,35,37,41,46,52,54,59,63,65,71,74,80,
%T A112045 88,91,93,97,99,103,116,120,126,128,137,139,145,151,155,160,166,168,
%U A112045 178,180,183,185,197,209,212,214,218,224,226,236,241,247,253,255,261
%N A112045 Position of primes (A000040) among square-free numbers A000037.
%Y A112045 a(n) = A028391(A000040(n)).
%K A112045 nonn
%O A112045 1,2
%A A112045 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112046
%S A112046 2,2,3,3,2,2,3,3,2,2,5,5,2,2,3,3,2,2,3,3,2,2,5,7,2,2,3,3,2,2,3,3,2,2,
%T A112046 7,5,2,2,3,3,2,2,3,3,2,2,5,5,2,2,3,3,2,2,3,3,2,2,7,11,2,2,3,3,2,2,3,3,
%U A112046 2,2,5,5,2,2,3,3,2,2,3,3,2,2,5,13,2,2,3,3,2,2,3,3,2,2,7,5,2,2,3,3,2,2
%N A112046 a(n) = The first i >= 1, for which Jacobi symbol J(i,2n+1) is not +1 (i.e. is either 0 or -1).
%C A112046 If we instead list the first i >= 1, for which Jacobi symbol J(i,2n+1) is 0, we get A090368.
%e A112046 It appears that every term is prime (A000040).
%Y A112046 a(n)=A112050(n)+1. Bisections: A112047, A112048. Their difference: A112053. C.f. A112049, A112060, A112070.
%K A112046 nonn
%O A112046 1,1
%A A112046 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112047
%S A112047 2,3,2,3,2,5,2,3,2,3,2,5,2,3,2,3,2,7,2,3,2,3,2,5,2,3,2,3,2,7,2,3,2,3,
%T A112047 2,5,2,3,2,3,2,5,2,3,2,3,2,7,2,3,2,3,2,5,2,3,2,3,2,7,2,3,2,3,2,5,2,3,
%U A112047 2,3,2,5,2,3,2,3,2,11,2,3,2,3,2,5,2,3,2,3,2,7,2,3,2,3,2,5,2,3,2,3,2,5
%N A112047 Bisection of A112046.
%F A112047 A112046(2n-1)
%Y A112047 C.f. A112048, A112053.
%K A112047 nonn
%O A112047 1,1
%A A112047 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112048
%S A112048 2,3,2,3,2,5,2,3,2,3,2,7,2,3,2,3,2,5,2,3,2,3,2,5,2,3,2,3,2,11,2,3,2,3,
%T A112048 2,5,2,3,2,3,2,13,2,3,2,3,2,5,2,3,2,3,2,5,2,3,2,3,2,7,2,3,2,3,2,5,2,3,
%U A112048 2,3,2,17,2,3,2,3,2,5,2,3,2,3,2,5,2,3,2,3,2,19,2,3,2,3,2,5,2,3,2,3,2
%N A112048 Bisection of A112046.
%F A112048 A112046(2n)
%Y A112048 C.f. A112047, A112053.
%K A112048 nonn
%O A112048 1,1
%A A112048 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112049
%S A112049 1,1,2,2,1,1,2,2,1,1,3,3,1,1,2,2,1,1,2,2,1,1,3,4,1,1,2,2,1,1,2,2,1,1,
%T A112049 4,3,1,1,2,2,1,1,2,2,1,1,3,3,1,1,2,2,1,1,2,2,1,1,4,5,1,1,2,2,1,1,2,2,
%U A112049 1,1,3,3,1,1,2,2,1,1,2,2,1,1,3,6,1,1,2,2,1,1,2,2,1,1,4,3,1,1,2,2,1,1
%N A112049 a(n) = Position of A112046(n) in A000040, 0 if not prime.
%F A112049 A049084(A112046(n))
%Y A112049 C.f. A112060.
%K A112049 nonn
%O A112049 1,3
%A A112049 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112050
%S A112050 1,1,2,2,1,1,2,2,1,1,4,4,1,1,2,2,1,1,2,2,1,1,4,6,1,1,2,2,1,1,2,2,1,1,
%T A112050 6,4,1,1,2,2,1,1,2,2,1,1,4,4,1,1,2,2,1,1,2,2,1,1,6,10,1,1,2,2,1,1,2,2,
%U A112050 1,1,4,4,1,1,2,2,1,1,2,2,1,1,4,12,1,1,2,2,1,1,2,2,1,1,6,4,1,1,2,2,1,1
%N A112050 The longest prefix of 1's in the Jacobi-vector {J(2n+1,1),J(2n+1,2),...,J(2n+1,2n)} of 2n+1.
%F A112050 a(n) = A112046(n)-1
%K A112050 nonn
%O A112050 1,3
%A A112050 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112051
%S A112051 1,3,11,24,60,84,144,180,264,420,480,684,840,924,1104,1404,1740,1860,
%T A112051 2244,2520,2664,3120,3444,3960,4704,5100,5304,5724,5940,6384,8064,
%U A112051 8580,9384,9660,11100,11400,12324,13284,13944,14964,16020,16380,18240
%N A112051 a(1)=1, a(n) = The first index i (> a(n-1)), where A112046(i) gets a value distinct from any values A112046(1)..A112046(a(n-1)).
%Y A112051 Column 1 of A112060 (row 1 of A112061). C.f. A112052.
%K A112051 nonn
%O A112051 1,2
%A A112051 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112052
%S A112052 3,7,23,49,121,169,289,361,529,841,961,1369,1681,1849,2209,2809,3481,
%T A112052 3721,4489,5041,5329,6241,6889,7921,9409,10201,10609,11449,11881,
%U A112052 12769,16129,17161,18769,19321,22201,22801,24649,26569,27889,29929
%N A112052 a(n) = 2*A112051(n)+1.
%F A112052 A005408(A112051(n))
%C A112052 From n>=4 onward seems to be squares of primes (A001248).
%Y A112052 Column 1 of A112070 (row 1 of A112071).
%K A112052 nonn
%O A112052 1,1
%A A112052 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112053
%S A112053 0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,
%T A112053 0,0,0,0,0,0,0,8,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A112053 0,0,0,12,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0
%V A112053 0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,
%W A112053 0,0,0,0,0,0,0,8,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%X A112053 0,0,0,12,0,0,0,0,0,-6,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0
%N A112053 A112048 - A112047.
%F A112053 a(n) = A112048(n)-A112047(n).
%Y A112053 Points where a(n) is not zero: A112054. Values at those points: A112059.
%K A112053 sign
%O A112053 1,12
%A A112053 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112054
%S A112054 12,18,30,42,48,72,78,90,102,108,120,132,138,162,168,180,192,198,210,
%T A112054 222,228,240,252,258,282,288,300,312,318,330,342,348,372,378,390,402,
%U A112054 408,420,432,438,450,462,468,492,498,510,522,528,540,552,558,582,588
%N A112054 Points where A112053 is not zero.
%C A112054 Apparently these are all divisible by 6. See A112055.
%Y A112054 A112058(n) = 4*a(n).
%K A112054 nonn
%O A112054 1,1
%A A112054 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112055
%S A112055 2,3,5,7,8,12,13,15,17,18,20,22,23,27,28,30,32,33,35,37,38,40,42,43,
%T A112055 47,48,50,52,53,55,57,58,62,63,65,67,68,70,72,73,75,77,78,82,83,85,87,
%U A112055 88,90,92,93,97,98,100,102,103,107,108,110,112,113,117,118,120,122
%N A112055 The numerator of A112054(n)/6.
%C A112055 All the terms of A112054 seem to be divisible by 6. If that is true, then this is a complement of A112082.
%K A112055 nonn
%O A112055 1,1
%A A112055 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112056
%S A112056 47,71,119,167,191,287,311,359,407,431,479,527,551,647,671,719,767,
%T A112056 791,839,887,911,959,1007,1031,1127,1151,1199,1247,1271,1319,1367,
%U A112056 1391,1487,1511,1559,1607,1631,1679,1727,1751,1799,1847,1871,1967
%N A112056 Odd numbers of the form 4n-1 for which jacobi-first-non-one(4n-1) differs from jacobi-first-non-one(4n+1).
%F A112056 a(n) = 4*A112054(n)-1.
%C A112056 Here jacobi-first-non-one(m) (for odd numbers m) is defined as the first value of i >= 1, for which Jacobi symbol J(i,m) is not +1 (i.e. is either 0 or -1).
%Y A112056 a(n) = A112057(n)-2 = A112058(n)-1.
%K A112056 nonn
%O A112056 1,1
%A A112056 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112057
%S A112057 49,73,121,169,193,289,313,361,409,433,481,529,553,649,673,721,769,
%T A112057 793,841,889,913,961,1009,1033,1129,1153,1201,1249,1273,1321,1369,
%U A112057 1393,1489,1513,1561,1609,1633,1681,1729,1753,1801,1849,1873,1969
%N A112057 Odd numbers of the form 4n+1 for which jacobi-first-non-one(4n-1) differs from jacobi-first-non-one(4n+1).
%F A112057 a(n) = 4*A112054(n)+1.
%C A112057 The definition of jacobi-first-non-one is given in A112056.
%Y A112057 a(n) = A112056(n)+2 = A112058(n)+1.
%K A112057 nonn
%O A112057 1,1
%A A112057 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112058
%S A112058 48,72,120,168,192,288,312,360,408,432,480,528,552,648,672,720,768,
%T A112058 792,840,888,912,960,1008,1032,1128,1152,1200,1248,1272,1320,1368,
%U A112058 1392,1488,1512,1560,1608,1632,1680,1728,1752,1800,1848,1872,1968
%N A112058 Mean of A112056 and A112057.
%F A112058 a(n) = 4*A112054(n).
%Y A112058 a(n) = A112056(n)+1 = A112057(n)-1 = (A112056(n)+A112057(n))/2.
%K A112058 nonn
%O A112058 1,1
%A A112058 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112059
%S A112059 2,2,4,8,2,12,6,12,2,2,6,18,6,2,6,4,2,2,18,2,2,24,6,2,6,8,
%T A112059 4,2,2,6,32,6,2,6,10,2,2,28,2,2,4,38,2,6,6,4,2,2,4,42,6,2,
%U A112059 8,4,2,2,2,2,6,8,2,48,6,4,2,2,10,6,12,2,6,4,2,2,2,2,52,8
%V A112059 2,-2,4,8,-2,12,-6,12,2,-2,-6,18,-6,2,-6,-4,2,-2,18,2,-2,24,6,-2,6,-8,
%W A112059 4,2,-2,-6,32,-6,2,-6,-10,2,-2,28,2,-2,4,38,-2,6,-6,4,2,-2,-4,42,-6,2,
%X A112059 -8,-4,2,-2,2,-2,6,8,-2,48,-6,4,2,-2,-10,6,-12,2,-6,-4,2,-2,2,-2,52,8
%N A112059 Non-zero terms of A112053 and A112080.
%Y A112059 a(n) = A112053(A112054(n)).
%K A112059 sign
%O A112059 1,1
%A A112059 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112060
%S A112060 1,2,3,5,4,11,6,7,12,24,9,8,23,35,60,10,15,36,59,155,84,13,16,47,95,
%T A112060 275,239,144,14,19,48,119,335,575,779,180,17,20,71,120,359,659,1499,
%U A112060 2855,264,18,27,72,179,419,839,1535,4199,5279,420,21,28,83,204,504
%N A112060 Square array A(col>=1, row>=1) by antidiagonals A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), ..., where A(x,y) contains y:th natural number i for which A112049(i)=x, and 0 if no such i exists.
%C A112060 This is a permutation of natural numbers provided that the sequence A112046 contains only prime values and every prime occurs infinitely many times there.
%e A112060 The top left corner of the array:
%e A112060 1,2,5,6,9,10,...
%e A112060 3,4,7,8,15,16,,...
%e A112060 11,12,23,36,47,...
%H A112060 Index entries for sequences that are permutations of the natural numbers
%Y A112060 A112070(x,y) = 2*A(X,Y)+1. Transpose: A112061. Column 1: A112051. Row 1: A042963, Row 2: A112062, Row 3: A112063, Row 4: A112064, Row 5: A112065, Row 6: A112066, Row 7: A112067, Row 8: A112068, Row 9: A112069.
%K A112060 nonn,tabl
%O A112060 1,2
%A A112060 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112061
%S A112061 1,3,2,11,4,5,24,12,7,6,60,35,23,8,9,84,155,59,36,15,10,144,239,275,
%T A112061 95,47,16,13,180,779,575,335,119,48,19,14,264,2855,1499,659,359,120,
%U A112061 71,20,17,420,5279,4199,1535,839,419,179,72,27,18,480,9095,7895,4619
%N A112061 Transpose of A112060.
%H A112061 Index entries for sequences that are permutations of the natural numbers
%K A112061 nonn,tabl
%O A112061 1,2
%A A112061 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112070
%S A112070 3,5,7,11,9,23,13,15,25,49,19,17,47,71,121,21,31,73,119,311,169,27,33,
%T A112070 95,191,551,479,289,29,39,97,239,671,1151,1559,361,35,41,143,241,719,
%U A112070 1319,2999,5711,529,37,55,145,359,839,1679,3071,8399,10559,841,43,57
%N A112070 Square array A(col>=1, row>=1) by antidiagonals A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), ..., where A(x,y) contains y:th odd number 2i+1 (i>=1) for which A112049(2i+1)=x, and 0 if no such i exists.
%C A112070 This is a permutation of odd numbers greater than unity provided that the sequence A112046 contains only prime values and every prime occurs infinitely many times there. Because the Jacobi symbol is multiplicative with respect to its modulus, it follows that if n occurs on row i, and m occurs on row j, then n*m cannot occur before row min(i,j).
%e A112070 The top left corner of the array:
%e A112070 3,5,11,13,19,21,...
%e A112070 7,9,15,17,31,33,,...
%e A112070 23,25,47,73,95,...
%Y A112070 A(x,y) = 2*A112060(x,y)+1. Transpose: A112071. Column 1: A112052.Row 1: A047621, Row 2: A112072 Row 3: A112073, Row 4: A112074, Row 5: A112075, Row 6: A112076, Row 7: A112077, Row 8: A112078, Row 9: A112079.
%K A112070 nonn,tabl
%O A112070 1,1
%A A112070 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112071
%S A112071 3,7,5,23,9,11,49,25,15,13,121,71,47,17,19,169,311,119,73,31,21,289,
%T A112071 479,551,191,95,33,27,361,1559,1151,671,239,97,39,29,529,5711,2999,
%U A112071 1319,719,241,143,41,35,841,10559,8399,3071,1679,839,359,145,55,37
%N A112071 Transpose of A112070.
%H A112071 Index entries for sequences that are permutations of the natural numbers
%K A112071 nonn,tabl
%O A112071 1,1
%A A112071 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112062
%S A112062 3,4,7,8,15,16,19,20,27,28,31,32,39,40,43,44,51,52,55,56,63,64,67,68,
%T A112062 75,76,79,80,87,88,91,92,99,100,103,104,111,112,115,116,123,124,127,
%U A112062 128,135,136,139,140,147,148,151,152,159,160,163,164,171,172,175,176
%N A112062 Positive integers i for which A112049(i) == 2
%Y A112062 Row 2 of A112060.
%K A112062 nonn
%O A112062 1,1
%A A112062 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112063
%S A112063 11,12,23,36,47,48,71,72,83,96,107,108,131,132,143,156,167,168,191,
%T A112063 192,203,216,227,228,251,252,263,276,287,288,311,312,323,336,347,348,
%U A112063 371,372,383,396,407,408,431,432,443,456,467,468,491,492,503,516,527
%N A112063 Positive integers i for which A112049(i) == 3
%Y A112063 Row 3 of A112060.
%K A112063 nonn
%O A112063 1,1
%A A112063 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112064
%S A112064 24,35,59,95,119,120,179,204,215,240,299,300,324,360,384,395,444,455,
%T A112064 479,515,539,540,599,624,635,660,719,720,744,780,804,815,864,875,899,
%U A112064 935,959,960,1019,1044,1055,1080,1139,1140,1164,1200,1224,1235,1284
%N A112064 Positive integers i for which A112049(i) == 4
%Y A112064 Row 4 of A112060.
%K A112064 nonn
%O A112064 1,1
%A A112064 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112065
%S A112065 60,155,275,335,359,419,504,564,600,695,755,900,984,995,1020,1079,
%T A112065 1115,1199,1259,1260,1415,1440,1524,1595,1619,1679,1680,1824,1919,
%U A112065 1944,2015,2039,2100,2184,2255,2280,2339,2364,2435,2519,2580,2604
%N A112065 Positive integers i for which A112049(i) == 5
%Y A112065 Row 5 of A112060.
%K A112065 nonn
%O A112065 1,1
%A A112065 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112066
%S A112066 84,239,575,659,839,1175,1320,1344,1764,1835,1955,2099,2160,2375,2459,
%T A112066 2759,2784,2879,3084,3299,3515,3695,3779,3780,3840,3935,4139,4439,
%U A112066 4475,4620,4764,4800,4859,4884,5040,5544,5795,5964,6024,6119,6155
%N A112066 Positive integers i for which A112049(i) == 6
%Y A112066 Row 6 of A112060.
%K A112066 nonn
%O A112066 1,1
%A A112066 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112067
%S A112067 144,779,1499,1535,3215,3275,4044,4355,4380,4535,5459,5460,6864,6995,
%T A112067 7284,7379,7499,7704,8315,8399,9059,9420,9504,10079,10164,10560,10775,
%U A112067 10980,11075,11699,11760,11999,12024,12455,12935,13715,13775,14040
%N A112067 Positive integers i for which A112049(i) == 7
%Y A112067 Row 7 of A112060.
%K A112067 nonn
%O A112067 1,1
%A A112067 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112068
%S A112068 180,2855,4199,4619,5195,5399,6719,7475,8555,9000,9815,10739,13859,
%T A112068 14340,16235,16319,16524,17159,18299,18564,18744,20015,20579,21359,
%U A112068 22320,22524,22619,23759,24275,24504,24960,25884,26544,27455,27720
%N A112068 Positive integers i for which A112049(i) == 8
%Y A112068 Row 8 of A112060.
%K A112068 nonn
%O A112068 1,1
%A A112068 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112069
%S A112069 264,5279,7895,17135,19979,25475,26615,26940,32579,34799,41435,42900,
%T A112069 43320,44039,44459,46019,46284,48959,51000,56939,58560,61595,64259,
%U A112069 64595,65364,69155,71135,72084,76295,78624,79379,82320,83159,83304
%N A112069 Positive integers i for which A112049(i) == 9
%Y A112069 Row 9 of A112060.
%K A112069 nonn
%O A112069 1,1
%A A112069 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112072
%S A112072 7,9,15,17,31,33,39,41,55,57,63,65,79,81,87,89,103,105,111,113,127,
%T A112072 129,135,137,151,153,159,161,175,177,183,185,199,201,207,209,223,225,
%U A112072 231,233,247,249,255,257,271,273,279,281,295,297,303,305,319,321,327
%N A112072 Odd numbers n for which 3 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.
%Y A112072 Row 2 of A112070. a(n) = 2*A112062(n)+1.
%K A112072 nonn
%O A112072 1,1
%A A112072 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112073
%S A112073 23,25,47,73,95,97,143,145,167,193,215,217,263,265,287,313,335,337,
%T A112073 383,385,407,433,455,457,503,505,527,553,575,577,623,625,647,673,695,
%U A112073 697,743,745,767,793,815,817,863,865,887,913,935,937,983,985,1007
%N A112073 Odd numbers n for which 5 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.
%Y A112073 Row 3 of A112070. a(n) = 2*A112063(n)+1.
%K A112073 nonn
%O A112073 1,1
%A A112073 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112074
%S A112074 49,71,119,191,239,241,359,409,431,481,599,601,649,721,769,791,889,
%T A112074 911,959,1031,1079,1081,1199,1249,1271,1321,1439,1441,1489,1561,1609,
%U A112074 1631,1729,1751,1799,1871,1919,1921,2039,2089,2111,2161,2279,2281
%N A112074 Odd numbers n for which 7 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.
%Y A112074 Row 4 of A112070. a(n) = 2*A112064(n)+1.
%K A112074 nonn
%O A112074 1,1
%A A112074 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112075
%S A112075 121,311,551,671,719,839,1009,1129,1201,1391,1511,1801,1969,1991,2041,
%T A112075 2159,2231,2399,2519,2521,2831,2881,3049,3191,3239,3359,3361,3649,
%U A112075 3839,3889,4031,4079,4201,4369,4511,4561,4679,4729,4871,5039,5161
%N A112075 Odd numbers n for which 11 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.
%Y A112075 Row 5 of A112070. a(n) = 2*A112065(n)+1.
%K A112075 nonn
%O A112075 1,1
%A A112075 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112076
%S A112076 169,479,1151,1319,1679,2351,2641,2689,3529,3671,3911,4199,4321,4751,
%T A112076 4919,5519,5569,5759,6169,6599,7031,7391,7559,7561,7681,7871,8279,
%U A112076 8879,8951,9241,9529,9601,9719,9769,10081,11089,11591,11929,12049
%N A112076 Odd numbers n for which 13 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.
%Y A112076 Row 6 of A112070. a(n) = 2*A112066(n)+1.
%K A112076 nonn
%O A112076 1,1
%A A112076 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112077
%S A112077 289,1559,2999,3071,6431,6551,8089,8711,8761,9071,10919,10921,13729,
%T A112077 13991,14569,14759,14999,15409,16631,16799,18119,18841,19009,20159,
%U A112077 20329,21121,21551,21961,22151,23399,23521,23999,24049,24911,25871
%N A112077 Odd numbers n for which 17 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.
%Y A112077 Row 7 of A112070. a(n) = 2*A112067(n)+1.
%K A112077 nonn
%O A112077 1,1
%A A112077 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112078
%S A112078 361,5711,8399,9239,10391,10799,13439,14951,17111,18001,19631,21479,
%T A112078 27719,28681,32471,32639,33049,34319,36599,37129,37489,40031,41159,
%U A112078 42719,44641,45049,45239,47519,48551,49009,49921,51769,53089,54911
%N A112078 Odd numbers n for which 19 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.
%Y A112078 Row 8 of A112070. a(n) = 2*A112068(n)+1.
%K A112078 nonn
%O A112078 1,1
%A A112078 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112079
%S A112079 529,10559,15791,34271,39959,50951,53231,53881,65159,69599,82871,
%T A112079 85801,86641,88079,88919,92039,92569,97919,102001,113879,117121,
%U A112079 123191,128519,129191,130729,138311,142271,144169,152591,157249
%N A112079 Odd numbers n for which 23 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.
%Y A112079 Row 9 of A112070. a(n) = 2*A112069(n)+1.
%K A112079 nonn
%O A112079 1,1
%A A112079 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112080
%S A112080 0,2,2,0,4,0,8,2,0,0,0,12,6,0,12,0,2,2,0,6,0,18,6,0,0,0,2,6,0,
%T A112080 4,0,2,2,0,18,0,2,2,0,24,0,6,2,0,0,0,6,8,0,4,0,2,2,0,6,0,32,6,
%U A112080 0,0,0,2,6,0,10,0,2,2,0,28,0,2,2,0,4,0,38,2,0,0,0,6,6,0,4,0,2
%V A112080 0,2,-2,0,4,0,8,-2,0,0,0,12,-6,0,12,0,2,-2,0,-6,0,18,-6,0,0,0,2,-6,0,
%W A112080 -4,0,2,-2,0,18,0,2,-2,0,24,0,6,-2,0,0,0,6,-8,0,4,0,2,-2,0,-6,0,32,-6,
%X A112080 0,0,0,2,-6,0,-10,0,2,-2,0,28,0,2,-2,0,4,0,38,-2,0,0,0,6,-6,0,4,0,2
%N A112080 a(n) = Difference between jacobi-first-non-one(24n+1) and jacobi-first-non-one(24n-1).
%F A112080 a(n) = 4*A112054(n)+1.
%C A112080 The definition of jacobi-first-non-one is given in A112056.
%Y A112080 A112081(n) = a(n)/2. A112082 gives the points where a(n) is zero. A112059 gives the non-zero terms.
%K A112080 sign
%O A112080 1,2
%A A112080 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112081
%S A112081 0,1,1,0,2,0,4,1,0,0,0,6,3,0,6,0,1,1,0,3,0,9,3,0,0,0,1,3,0,2,
%T A112081 0,1,1,0,9,0,1,1,0,12,0,3,1,0,0,0,3,4,0,2,0,1,1,0,3,0,16,3,0,0,
%U A112081 0,1,3,0,5,0,1,1,0,14,0,1,1,0,2,0,19,1,0,0,0,3,3,0,2,0,1,1,0
%V A112081 0,1,-1,0,2,0,4,-1,0,0,0,6,-3,0,6,0,1,-1,0,-3,0,9,-3,0,0,0,1,-3,0,-2,
%W A112081 0,1,-1,0,9,0,1,-1,0,12,0,3,-1,0,0,0,3,-4,0,2,0,1,-1,0,-3,0,16,-3,0,0,
%X A112081 0,1,-3,0,-5,0,1,-1,0,14,0,1,-1,0,2,0,19,-1,0,0,0,3,-3,0,2,0,1,-1,0
%N A112081 A112080(n)/2.
%K A112081 sign
%O A112081 1,5
%A A112081 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A112082
%S A112082 1,4,6,9,10,11,14,16,19,21,24,25,26,29,31,34,36,39,41,44,45,46,49,51,
%T A112082 54,56,59,60,61,64,66,69,71,74,76,79,80,81,84,86,89,91,94,95,96,99,
%U A112082 101,104,105,106,109,111,114,115,116,119,121,124,126,129,130,131,134
%N A112082 Points where A112080 is zero.
%Y A112082 C.f. A112055.
%K A112082 nonn
%O A112082 1,2
%A A112082 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
-------------------------------------------------------------------------------
Yours,
Antti Karttunen