A239612 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).

1, 8, 30, 112, 220, 240, 546, 1280, 1134, 1760, 2310, 3360, 4212, 4368, 6600, 13312, 9520, 9072, 12654, 24640, 16380, 18480, 22770, 38400, 42500, 33696, 39366, 61152, 47908, 52800, 56730, 131072, 69300, 76160, 120120, 127008, 99900, 101232, 126360, 281600

Offset: 1

Author

José María Grau Ribas, Jun 25 2014

Keywords

• nonn
• mult

Comments

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• C. Calderón, J. M. Grau, A. Oller-Marcen, L. Toth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Crossrefs

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

Programs

Mathematica

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]

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

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

Extensions

Keyword:mult added by Andrew Howroyd, Aug 07 2018