A112054 Indices where A112053 is not zero.

12, 18, 30, 42, 48, 72, 78, 90, 102, 108, 120, 132, 138, 162, 168, 180, 192, 198, 210, 222, 228, 240, 252, 258, 282, 288, 300, 312, 318, 330, 342, 348, 372, 378, 390, 402, 408, 420, 432, 438, 450, 462, 468, 492, 498, 510, 522, 528, 540, 552, 558, 582, 588

Offset: 1

Author

Antti Karttunen, Aug 27 2005

Keywords

• nonn

Comments

These are all divisible by 6, as J(2,m) = +1 if m = 1 or 7 mod 8 and -1 if m = 3 or 5 mod 8 and J(3,m) = +1 if m = 1 or 11 mod 12, -1 if m = 5 or 7 mod 12 and 0 if m = 3 or 9 mod 12 (where Jacobi symbol J(i,m) returns +1 if i is quadratic residue modulo odd number m), it follows that only when i=24*n it holds that J(2,i-1)=J(2,i+1)=J(3,i-1)=J(3,i+1)=+1 and thus only then the function A112046 (and A112053) depends on values of J(k>3,m).

Crossrefs

Cf. A112058(n) = 4*a(n). A112055(n) = a(n)/6.

Programs

Mathematica

a112046[n_]:=Block[{i=1}, While[JacobiSymbol[i, 2n + 1]==1, i++]; i]; Select[Range[1000], a112046[2#] - a112046[2# - 1] != 0 &] (* Indranil Ghosh, May 24 2017 *)

Python

from sympy import jacobi_symbol as J
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a112046(2*n) - a112046(2*n - 1)
print([n for n in range(1, 1001) if a(n)!=0]) # Indranil Ghosh, May 24 2017