A165602 -id:A165602 - cp's OEIS frontend

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.

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

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.

A166091 Square array A(row>=0, col>=0) = (A166092(row,col)-3)/4, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 1, 2, 3, 79, 4, 5, 151, 8, 103, 7, 175, 14, 409, 22, 9, 223, 15, 589, 43, 10, 11, 265, 25, 998, 143, 31, 12, 13, 275, 32, 1297, 157, 73, 20, 16, 17, 283, 62, 1364, 182, 158, 55, 28, 6, 19, 361, 69, 1891, 293, 164, 183, 34, 26, 52, 21, 373, 74, 1952, 397, 401

Offset: 0

Antti Karttunen, Oct 08 2009

Note: This is not a permutation of nonnegative integers, as for some odd n, A166040(n) gets even value, the first example being A166040(49)=32, thus 24 (= (49-1)/2) is missing from here, and correspondingly, 99 (= 2*49 + 1) is missing from A166092. Sequence A165602 gives the natural numbers missing from this table.

			The top left corner of the array:
0, 1, 3, 5, 7, 9, ...
2, 79, 151, 175, 223, 265, ...
4, 8, 14, 15, 25, 32, ...
103, 409, 589, 998, 1297, 1364, ...
22, 43, 143, 157, 182, 293, ...

A. Karttunen, Table of n, a(n) for n = 0..10010

See A166092. The leftmost column: A166094. The first five rows: A165468, A166052, A166054, A166056, A166058. Cf. also A112060.

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.

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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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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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A166091 Square array A(row>=0, col>=0) = (A166092(row,col)-3)/4, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

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

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