A239613 a(n) = Sum_{0 < x,y,z,t <= n and gcd(x^2 + y^2 + z^2 + t^2, n)=1} gcd(x^2 + y^2 + z^2 + t^2 - 1, n).

1, 16, 96, 384, 960, 1536, 4032, 8192, 11664, 15360, 26400, 36864, 52416, 64512, 92160, 163840, 156672, 186624, 246240, 368640, 387072, 422400, 534336, 786432, 900000, 838656, 1259712, 1548288, 1364160, 1474560, 1785600, 3145728

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, A239612, A239614, A239615, A079458, A053191, A227499.

Programs

Mathematica

g4[n_] := Sum[If[GCD[x^2 + y^2+ z^2+ t^2, n] == 1, GCD[x^2 + y^2+ z^2+ t^2 - 1, n], 0], {x, 1, n}, {y, 1, n},{z,1,n},{t,1,n}]; Array[g4,100]

PARI

a(n) = {s = 0; for (x=1, n, for (y=1, n, for (z=1, n, for (t=1, n, if (gcd(x^2+y^2+z^2+t^2,n) == 1, s += gcd(x^2+y^2+z^2+t^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)^4)); 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