A239612 - OEIS

%I #22 Aug 08 2018 02:37:22

%S 1,8,30,112,220,240,546,1280,1134,1760,2310,3360,4212,4368,6600,13312,

%T 9520,9072,12654,24640,16380,18480,22770,38400,42500,33696,39366,

%U 61152,47908,52800,56730,131072,69300,76160,120120,127008,99900,101232,126360,281600

%N a(n) = Sum_{0 < x,y,z <= n and gcd(x^2 + y^2 + z^2, n)=1} gcd(x^2 + y^2 + z^2 - 1, n).

%C Related to Menon's identity. See Conclusions and further work section of the arXiv file linked.

%H Andrew Howroyd, <a href="/A239612/b239612.txt">Table of n, a(n) for n = 1..1000</a>

%H C. Calderón, J. M. Grau, A. Oller-Marcen, L. Toth, <a href="http://arxiv.org/abs/1403.7878">Counting invertible sums of squares modulo n and a new generalization of Euler totient function</a>, arXiv:1403.7878 [math.NT], 2014.

%t g3[n_] := Sum[If[GCD[x^2 + y^2 + z^2, n] == 1, GCD[x^2 + y^2 + z^2 - 1, n], 0],{x, 1, n},{y, 1, n},{z,1,n}]; Array[g3,100]

%o (PARI) a(n) = {s = 0; for (x=1, n, for (y=1, n, for (z=1, n, if (gcd(x^2+y^2+z^2,n) == 1, s += gcd(x^2+y^2+z^2-1,n));););); s;} \\ _Michel Marcus_, Jun 29 2014

%o (PARI) a(n)={my(p=lift(Mod(sum(i=0, n-1, x^(i^2%n)), x^n-1)^3)); sum(i=0, n-1, if(gcd(i,n)==1, polcoeff(p,i)*gcd((i-1)%n,n)))} \\ _Andrew Howroyd_, Aug 07 2018

%Y Cf. A239611, A239613, A239614, A239615, A079458, A053191, A227499.

%K nonn,mult

%O 1,2

%A _José María Grau Ribas_, Jun 25 2014

%E Keyword:mult added by _Andrew Howroyd_, Aug 07 2018