A165898 -id:A165898 - cp's OEIS frontend

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

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

A166405 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

0, 2, 5, 14, 0, 33, 39, 45, 68, 95, 63, 161, 0, 126, 203, 279, 165, 245, 333, 312, 410, 473, 270, 658, 0, 459, 689, 660, 513, 944, 915, 630, 780, 1139, 759, 1491, 1314, 775, 1155, 1738, 0, 1826, 1360, 1479, 1958, 1729, 1395, 2090, 2328, 1485, 2525, 2884

Offset: 0

Antti Karttunen, Oct 21 2009

nonn

			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) obtaining value -1 when i=2, 6, 7, 8 and 10, thus a(5)=33.

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

A125615(n)=a(A102781(n)). Cf. A166100, A166406-A166408. The cases where a(i)/A005408(i) is not integer seem also to be given by A166101. This is NOT a bisection of A165898. Scheme-code for jacobi-symbol is given at A165601.

Table[Total@ Select[Range[2n + 1], JacobiSymbol[#, 2n + 1]==-1 &], {n, 0, 100}] (* Indranil Ghosh, Jun 12 2017 *)

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

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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