A165605 -id:A165605 - cp's OEIS frontend

A166273 Bisection 2n+1 of A165605.

3, 3, 3, 6, 6, 9, 3, 6, 9, 6, 12, 9, 9, 9, 12, 9, 6, 6, 18, 0, 9, 15, 9, 15, 12, 15, 9, 12, 15, 12, 15, 12, 15, 21, 18, 21, 9, 9, 0, 12, 24, 12, 12, 15, 6, 18, 15, 18, 21, 24, 18, 12, 18, 27, 18, 24, 18, 12, 12, 12, 33, 21, 12, 33, 12, 27, 18, 21, 24, 6, 33, 15, 15, 24, 24, 30

Offset: 0

Antti Karttunen, Oct 13 2009

nonn

Conjecture: a(n) = -3 * A166269(n).

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

a(n) = A165605(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

A165460 The height at the 1/3 point of Jacobi-bridge, computed for 12n+7. a(n) = Sum_{i=0..(4n+2)} J(i,12n+7), where J(i,m) is the Jacobi symbol.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Conjecture: a(2n) = 2*A165605(2n) and a(2n+1) = (2/3)*A165605(2n+1). - Antti Karttunen, Oct 05 2009. (If true, then implies also the truth of conjecture in A165462.)

A. Karttunen, Table of n, a(n) for n = 0..21845

Cf. A165461, A165462, A165463, A165605.

Table[Sum[JacobiSymbol[i, 12n + 7], {i, 0, 4n + 2}], {n, 0, 100}] (* Indranil Ghosh, May 13 2017 *)

a(n) = sum(i=0, 4*n + 2, kronecker(i, 12*n + 7)); \\ Indranil Ghosh, May 13 2017

from sympy import jacobi_symbol as J
def a(n): return sum([J(i, 12*n + 7) for i in range(4*n + 3)]) # Indranil Ghosh, May 13 2017

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.

A165461 Positions of zeros in A165460.

Original entry on oeis.org

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.

A166268 Bisection 2n of A166050.

Original entry on oeis.org

1, 3, 4, 5, 5, 5, 7, 2, 9, 7, 6, 11, 8, 10, 1, 9, 14, 10, 15, 7, 7, 14, 14, 16, 8, 13, 13, 12, 18, 14, 13, 15, 0, 16, 9, 0, 22, 16, 19, 16, 11, 17, 16, 23, 19, 9, 22, 18, 16, 2, 18, 27, 12, 23, 11, 15, 24, 4, 27, 9, 23, 23, 20, 0, 19, 15, 22, 24, 22, 17, 18, 28, 26, 27, 17, 12

Offset: 0

Antti Karttunen, Oct 13 2009

nonn

Conjecture: a(n) = A166270(n)/2, and also A165605(A005843(n)).

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

a(n) = A166050(A005843(n)).

cp's OEIS Frontend

A166273 Bisection 2n+1 of A165605.

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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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A165460 The height at the 1/3 point of Jacobi-bridge, computed for 12n+7. a(n) = Sum_{i=0..(4n+2)} J(i,12n+7), where J(i,m) is the Jacobi symbol.

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

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A165461 Positions of zeros in A165460.

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A166268 Bisection 2n of A166050.

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