A166405 -id:A166405 - cp's OEIS frontend

A166406 a(n) = A166405(n)-A166100(n).

-1, 1, 0, 7, -27, 11, 0, 30, 0, 19, 0, 69, -250, 9, 0, 93, 0, 70, 0, 156, 0, 43, 0, 235, -1029, 102, 0, 220, 0, 177, 0, 126, 0, 67, 0, 497, 0, 50, 0, 395, -2187, 249, 0, 522, 0, 182, 0, 760, 0, 0, 0, 515, 0, 321, 0, 888, 0, 230, 0, 1190, -6655, 246, 0, 635, 0, 655, 0

Offset: 0

Antti Karttunen, Oct 21 2009, Oct 22 2009

sign

Zeros occur at (A166409(k)-1)/2. The negative terms occur at positions given by A046092 (see the comment at A166040).

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

Antti Karttunen, Table of n, a(n) for n = 0..65535

A125615(n)=a(A102781(n)). Cf. A166100, A166407-A166409. The cases where a(i)/A005408(i) is not integer seem also to be given by A166101.

from sympy import jacobi_symbol as J
def a(n):
    l=0
    m=0
    for i in range(1, 2*n + 2):
        if J(i, 2*n + 1)==-1: l+=i
        elif J(i, 2*n + 1)==1: m+=i
    return l - m
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 12 2017

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

A166406 a(n) = A166405(n)-A166100(n).

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