A166408 -id:A166408 - cp's OEIS frontend

A166405 Sum of those positive i <= 2n+1, for which J(i,2n+1)=-1. Here J(i,k) is the Jacobi symbol.

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

A166407 a(n) = floor(3*(A166406(n)/A005408(n))).

Original entry on oeis.org

-3, 1, 0, 3, -9, 3, 0, 6, 0, 3, 0, 9, -30, 1, 0, 9, 0, 6, 0, 12, 0, 3, 0, 15, -63, 6, 0, 12, 0, 9, 0, 6, 0, 3, 0, 21, 0, 2, 0, 15, -81, 9, 0, 18, 0, 6, 0, 24, 0, 0, 0, 15, 0, 9, 0, 24, 0, 6, 0, 30, -165, 6, 0, 15, 0, 15, 0, 6, 0, 9, 0, 30, 0, 0, 0, 21, 0, 12, 0, 30, 0, 3, 0, 33, -234, 6, 0, 6

Offset: 0

Antti Karttunen, Oct 21 2009

sign

Conjecture: the quotient A166406(i)/A005408(i) has denominator 3 when i is one of the terms of A166101, and it is integral in other cases. If true, then floor in the formula is unnecessary.

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

Cf. A166408.

from sympy import floor, 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 floor(3*((l - m)/(2*n + 1)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 12 2017

cp's OEIS Frontend

A166405 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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