A112054 - OEIS

%I #11 Apr 11 2021 04:08:38

%S 12,18,30,42,48,72,78,90,102,108,120,132,138,162,168,180,192,198,210,

%T 222,228,240,252,258,282,288,300,312,318,330,342,348,372,378,390,402,

%U 408,420,432,438,450,462,468,492,498,510,522,528,540,552,558,582,588

%N Indices where A112053 is not zero.

%C 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).

%t 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 *)

%o (Python)

%o from sympy import jacobi_symbol as J

%o def a112046(n):

%o     i=1

%o     while True:

%o         if J(i, 2*n + 1)!=1: return i

%o         else: i+=1

%o def a(n): return a112046(2*n) - a112046(2*n - 1)

%o print([n for n in range(1, 1001) if a(n)!=0]) # _Indranil Ghosh_, May 24 2017

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

%K nonn

%O 1,1

%A _Antti Karttunen_, Aug 27 2005