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

0,3

Comments

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.

Links

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

Wikipedia, Jacobi symbol

Example

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.

Mathematica

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

Prog

(MIT Scheme:) (define (A166100 n) (let ((w (A005408 n))) (let loop ((i 1) (s 1)) (cond ((= i w) s) (else (loop (1+ i) (+ s (if (= 1 (jacobi-symbol (1+ i) w)) (1+ i) 0))))))))
(Python)
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

Crossrefs

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

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

Sequence in context: A067701 A059240 A324363 * A135606 A051845 A278646

Adjacent sequences: A166097 A166098 A166099 * A166101 A166102 A166103

Keyword

nonn

Author

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

Status

approved