A165601 -id:A165601 - cp's OEIS frontend

A165605 Trisection 3n+1 of A165601.

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)).

A165602 Positions of zeros in A165601.

Original entry on oeis.org

24, 36, 51, 68, 78, 105, 118, 126, 132, 159, 186, 193, 211, 213, 222, 232, 240, 243, 267, 270, 294, 318, 321, 330, 348, 368, 375, 379, 402, 429, 443, 456, 465, 483, 493, 510, 537, 564, 568, 574, 575, 591, 618, 645, 672, 673, 693, 699, 708, 720, 722, 726

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

A. Karttunen, Table of n, a(n) for n = 0..21422

Cf. A165603, A165462.

A166045 Positions of records in A165601.

Original entry on oeis.org

0, 2, 8, 14, 29, 32, 62, 74, 104, 152, 164, 182, 224, 242, 272, 419, 422, 434, 452, 554, 662, 734, 812, 1022, 1064, 1154, 1274, 1442, 1742, 1904, 2072, 2132, 2384, 2762, 3002, 3464, 3584, 3884, 4712, 4802, 4844, 5114, 5642, 6272, 6362, 6542, 6692

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

A. Karttunen, Table of n, a(n) for n = 0..120

See A165466 and A166047 for the records themselves.

A166047 Record values in A165601.

Original entry on oeis.org

1, 3, 6, 9, 10, 15, 21, 24, 27, 30, 33, 36, 42, 45, 51, 52, 54, 60, 69, 72, 78, 87, 93, 99, 105, 108, 117, 132, 135, 138, 141, 150, 165, 186, 195, 201, 210, 216, 222, 225, 243, 246, 261, 264, 270, 273, 294, 297, 309, 312, 324, 342, 348, 354, 378, 390, 393, 399

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

A. Karttunen, Table of n, a(n) for n = 0..120

a(n) = A165601(A166045(n)). See A166046 for the corresponding 4k+3 integers.

A166046 The 4k+3 integers corresponding to the record positions in A165601.

Original entry on oeis.org

3, 11, 35, 59, 119, 131, 251, 299, 419, 611, 659, 731, 899, 971, 1091, 1679, 1691, 1739, 1811, 2219, 2651, 2939, 3251, 4091, 4259, 4619, 5099, 5771, 6971, 7619, 8291, 8531, 9539, 11051, 12011, 13859, 14339, 15539, 18851, 19211, 19379, 20459

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

A. Karttunen, Table of n, a(n) for n = 0..120

a(n) = A004767(A166045(n)). See A166047 for the records themselves.

A165604 Trisection 3n of A165601.

Original entry on oeis.org

1, 2, 1, 4, 6, 2, 2, 6, 0, 8, 6, 2, 0, 10, 6, 8, 12, 0, 12, 12, 1, 12, 6, 6, 12, 10, 0, 12, 18, 4, 2, 2, 6, 16, 18, 0, 12, 14, 6, 16, 12, 8, 0, 18, 0, 12, 12, 2, 24, 22, 6, 20, 12, 0, 24, 16, 2, 12, 30, 10, 12, 4, 0, 24, 18, 6, 12, 14, 12, 28, 18, 0, 2, 22, 0, 16, 24, 10, 24, 26, 0, 0

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

a(n) = A165601(A008585(n)).

A165606 Trisection 3n+2 of A165601.

Original entry on oeis.org

3, 3, 6, 5, 9, 7, 9, 8, 9, 10, 15, 10, 12, 11, 15, 13, 12, 14, 15, 15, 21, 13, 0, 14, 24, 19, 12, 18, 15, 19, 24, 17, 24, 16, 27, 21, 15, 20, 21, 25, 27, 21, 18, 18, 6, 26, 27, 6, 21, 25, 30, 22, 30, 23, 33, 30, 15, 24, 18, 31, 36, 21, 36, 22, 30, 32, 30, 30, 21, 33, 30, 21

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

a(n) = A165601(A016789(n)).

A166040 Number of times Sum_{i=1..u} J(i,2n+1) obtains value zero when u ranges from 1 to (2n+1). Here J(i,k) is the Jacobi symbol.

Original entry on oeis.org

0, 1, 2, 1, 0, 3, 4, 1, 4, 5, 6, 1, 0, 17, 8, 1, 4, 5, 8, 1, 8, 11, 20, 1, 0, 13, 14, 1, 6, 5, 10, 5, 8, 15, 14, 1, 8, 29, 20, 1, 0, 13, 10, 1, 14, 9, 20, 1, 8, 32, 24, 5, 12, 17, 12, 1, 14, 15, 38, 1, 0, 37, 74, 11, 10, 5, 18, 17, 12, 15, 22, 1, 10, 90, 22, 1, 38, 17, 22, 1, 14, 27, 18

Offset: 0

Antti Karttunen, Oct 08 2009

nonn

A046092 gives the positions of zeros, as only with odd squares A016754(m) = A005408(A046092(m)) Jacobi symbols J(i,n) never obtain value -1, and thus their partial sum never descends back to zero. Even positions contain only even values, while odd positions contain odd values in all other positions, except even values in the positions given by A005408(A165602(i)), for i>=0.

Four bold conjectures by Antti Karttunen, Oct 08 2009: 1) All odd natural numbers occur. 2) Each of them occurs infinitely many times. 3) All even natural numbers occur. 4) Each even number > 0 occurs only finitely many times. (The last can be disputed. For example, 6 occurs four times among the first 400001 terms, at the positions 10, 28, 360, 215832.)

Antti Karttunen, Table of n, a(n) for n = 0..400000

Bisections: A166085, A166086. See also A166087, A165601, A166092.

A165603 Numbers of the form 4n+3 for which Sum_{i=0..(2n+1)} J(i,4n+3) = 0, where J(i,m) is the Jacobi symbol.

Original entry on oeis.org

99, 147, 207, 275, 315, 423, 475, 507, 531, 639, 747, 775, 847, 855, 891, 931, 963, 975, 1071, 1083, 1179, 1275, 1287, 1323, 1395, 1475, 1503, 1519, 1611, 1719, 1775, 1827, 1863, 1935, 1975, 2043, 2151, 2259, 2275, 2299, 2303, 2367, 2475, 2583

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Integers of type 4n+3 whose midpoint height of Jacobi-bridge (A165601) is zero.

A. Karttunen, Table of n, a(n) for n = 0..21422

a(n) = A004767(A165602(n)). Cf. A165463, A165603, A165608.

A166100 Sum of those positive i <= 2n+1, for which J(i,2n+1)=+1. Here J(i,k) is the Jacobi symbol.

Original entry on oeis.org

1, 1, 5, 7, 27, 22, 39, 15, 68, 76, 63, 92, 250, 117, 203, 186, 165, 175, 333, 156, 410, 430, 270, 423, 1029, 357, 689, 440, 513, 767, 915, 504, 780, 1072, 759, 994, 1314, 725, 1155, 1343, 2187, 1577, 1360, 957, 1958, 1547, 1395, 1330, 2328, 1485, 2525

Offset: 0

Antti Karttunen, Oct 13 2009. Erroneous name corrected Oct 20 2009

nonn

Note that this sequence is not equal to the sum of the quadratic residues of 2n+1 in range [1,2n+1], and thus NOT a bisection of A165898.

			For n=5, we get odd number 11 (2*5+1), and J(i,11) = 1,-1,1,1,1,-1,-1,-1,1,-1,0 when i ranges from 1 to 11, J(i,11) getting value 1 when i=1, 3, 4, 5 and 9, thus a(5)=22.

A. Karttunen, Table of n, a(n) for n = 0..131071
Wikipedia, Jacobi symbol

A076409(n)=a(A102781(n)). Cf. A166101-A166104, A166405, A166406.

Scheme-code for jacobi-symbol is given at A165601.

Table[Total[Flatten[Position[JacobiSymbol[Range[2n+1],2n+1],1]]],{n,0,50}] (* Harvey P. Dale, Jun 19 2013 *)

from sympy import jacobi_symbol as J
def a(n): return sum([i for i in range(1, 2*n + 2) if J(i, 2*n + 1)==1]) # Indranil Ghosh, Jun 12 2017

cp's OEIS Frontend

A165605 Trisection 3n+1 of A165601.

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A165602 Positions of zeros in A165601.

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A166045 Positions of records in A165601.

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A166047 Record values in A165601.

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A166046 The 4k+3 integers corresponding to the record positions in A165601.

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A165604 Trisection 3n of A165601.

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A165606 Trisection 3n+2 of A165601.

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A166040 Number of times Sum_{i=1..u} J(i,2n+1) obtains value zero when u ranges from 1 to (2n+1). Here J(i,k) is the Jacobi symbol.

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A165603 Numbers of the form 4n+3 for which Sum_{i=0..(2n+1)} J(i,4n+3) = 0, where J(i,m) is the Jacobi symbol.

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A166100 Sum of those positive i <= 2n+1, for which J(i,2n+1)=+1. Here J(i,k) is the Jacobi symbol.

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