A165460 -id:A165460 - cp's OEIS frontend

A165461 Positions of zeros in A165460.

39, 64, 70, 77, 126, 164, 189, 191, 224, 289, 308, 314, 414, 420, 433, 439, 457, 469, 490, 539, 564, 567, 605, 664, 689, 763, 774, 789, 796, 798, 812, 814, 837, 910, 914, 932, 939, 989, 1035, 1039, 1064, 1078, 1106, 1112, 1155, 1164, 1189, 1253, 1280

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Also the positions of zeros in A165605 if the conjecture given in A165462 is true.

A. Karttunen, Table of n, a(n) for n = 0..1000

Cf. A165462-A165463.

A166270 Bisection 2n of A165460.

Original entry on oeis.org

2, 6, 8, 10, 10, 10, 14, 4, 18, 14, 12, 22, 16, 20, 2, 18, 28, 20, 30, 14, 14, 28, 28, 32, 16, 26, 26, 24, 36, 28, 26, 30, 0, 32, 18, 0, 44, 32, 38, 32, 22, 34, 32, 46, 38, 18, 44, 36, 32, 4, 36, 54, 24, 46, 22, 30, 48, 8, 54, 18, 46, 46, 40, 0, 38, 30, 44, 48, 44, 34, 36

Offset: 0

Antti Karttunen, Oct 13 2009

nonn

Conjecture: a(n) = 2*A166268(n).

A. Karttunen, Table of n, a(n) for n = 0..16383

a(n) = A165460(A005843(n)).

A166271 Bisection 2n+1 of A165460.

Original entry on oeis.org

2, 2, 2, 4, 4, 6, 2, 4, 6, 4, 8, 6, 6, 6, 8, 6, 4, 4, 12, 0, 6, 10, 6, 10, 8, 10, 6, 8, 10, 8, 10, 8, 10, 14, 12, 14, 6, 6, 0, 8, 16, 8, 8, 10, 4, 12, 10, 12, 14, 16, 12, 8, 12, 18, 12, 16, 12, 8, 8, 8, 22, 14, 8, 22, 8, 18, 12, 14, 16, 4, 22, 10, 10, 16, 16, 20, 12, 10, 20, 12, 16, 12

Offset: 0

Antti Karttunen, Oct 13 2009

nonn

Conjecture: a(n) = -2 * A166269(n).

A. Karttunen, Table of n, a(n) for n = 0..16383

a(n) = A165460(A005408(n)).

A165601 Midpoint height of Jacobi-bridge, computed for 4n+3. a(n) = Sum_{i=0..(2n+1)} J(i,4n+3), where J(i,m) is the Jacobi symbol.

Original entry on oeis.org

1, 1, 3, 2, 3, 3, 1, 3, 6, 4, 3, 5, 6, 4, 9, 2, 3, 7, 2, 5, 9, 6, 6, 8, 0, 5, 9, 8, 6, 10, 6, 5, 15, 2, 9, 10, 0, 7, 12, 10, 3, 11, 6, 2, 15, 8, 6, 13, 12, 9, 12, 0, 9, 14, 12, 7, 15, 12, 6, 15, 1, 6, 21, 12, 12, 13, 6, 11, 0, 6, 9, 14, 12, 8, 24, 10, 9, 19, 0, 10, 12, 12, 9, 18, 18, 1, 15

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

A. Karttunen, Table of n, a(n) for n = 0..269535

Trisections: A165604, A165605, A165606.

Cf. A165602, A165603, A165460, A166045, A166046, A166047.

Table[Sum[JacobiSymbol[i, 4n + 3], {i, 0, 2n + 1}], {n, 0, 100}] (* Indranil Ghosh, May 13 2017 *)

a(n) = sum(i=0, 2*n + 1, kronecker(i, 4*n + 3)); \\ Indranil Ghosh, May 13 2017

from sympy import jacobi_symbol as J
def a(n): return sum([J(i, 4*n + 3) for i in range(2*n + 2)]) # Indranil Ghosh, May 13 2017

A165605 Trisection 3n+1 of A165601.

Original entry on oeis.org

1, 3, 3, 3, 4, 3, 5, 6, 5, 6, 5, 9, 7, 3, 2, 6, 9, 9, 7, 6, 6, 12, 11, 9, 8, 9, 10, 9, 1, 12, 9, 9, 14, 6, 10, 6, 15, 18, 7, 0, 7, 9, 14, 15, 14, 9, 16, 15, 8, 12, 13, 15, 13, 9, 12, 12, 18, 15, 14, 12, 13, 15, 15, 12, 0, 15, 16, 21, 9, 18, 0, 21, 22, 9, 16, 9, 19, 0, 16, 12, 11, 24, 17

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Positions of zeros probably given by A165461. See the conjectures in A165460 and A165462.

A. Karttunen, Table of n, a(n) for n = 0..89844

a(n) = A165601(A016777(n)).

A165462 a(n) = (A165463(n)-3)/4.

Original entry on oeis.org

118, 193, 211, 232, 379, 493, 568, 574, 673, 868, 925, 943, 1243, 1261, 1300, 1318, 1372, 1408, 1471, 1618, 1693, 1702, 1816, 1993, 2068, 2290, 2323, 2368, 2389, 2395, 2437, 2443, 2512, 2731, 2743, 2797, 2818, 2968, 3106, 3118, 3193, 3235

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Conjecture: These are all those terms of A165602 which = 1 modulo 3. If this is true, then A165461 gives also the positions of zeros in A165605. - Antti Karttunen, Oct 05 2009

A. Karttunen, Table of n, a(n) for n = 0..1000

Cf. A165461-A165463. See also the conjecture in A165460.

A165463 Numbers of the form 12n+7 for which Sum_{i=0..(4n+2)} J(i,12n+7) = 0, where J(i,m) is the Jacobi symbol.

Original entry on oeis.org

475, 775, 847, 931, 1519, 1975, 2275, 2299, 2695, 3475, 3703, 3775, 4975, 5047, 5203, 5275, 5491, 5635, 5887, 6475, 6775, 6811, 7267, 7975, 8275, 9163, 9295, 9475, 9559, 9583, 9751, 9775, 10051, 10927, 10975, 11191, 11275, 11875, 12427

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Conjecture: a subset of A165603. See conjectures in A165460 and A165462.

A. Karttunen, Table of n, a(n) for n = 0..1000

a(n) = A004767(A165462(n)).

cp's OEIS Frontend

A165461 Positions of zeros in A165460.

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A166270 Bisection 2n of A165460.

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A166271 Bisection 2n+1 of A165460.

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A165601 Midpoint height of Jacobi-bridge, computed for 4n+3. a(n) = Sum_{i=0..(2n+1)} J(i,4n+3), where J(i,m) is the Jacobi symbol.

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A165605 Trisection 3n+1 of A165601.

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A165462 a(n) = (A165463(n)-3)/4.

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A165463 Numbers of the form 12n+7 for which Sum_{i=0..(4n+2)} J(i,12n+7) = 0, where J(i,m) is the Jacobi symbol.

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