A239611 a(n) = Sum_{0 < x,y <= n and gcd(x^2 + y^2, n)=1} gcd(x^2 + y^2 - 1, n).

1, 4, 16, 32, 32, 64, 96, 192, 216, 128, 240, 512, 288, 384, 512, 1024, 512, 864, 720, 1024, 1536, 960, 1056, 3072, 1200, 1152, 2592, 3072, 1568, 2048, 1920, 5120, 3840, 2048, 3072, 6912, 2592, 2880, 4608, 6144, 3200, 6144, 3696, 7680, 6912, 4224, 4416

Offset: 1

Author

José María Grau Ribas, Jun 24 2014

Keywords

• nonn
• mult

Comments

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

Multiplicative by the Chinese remainder theorem since gcd(x, m*n) = gcd(x, m)*gcd(x, n) for gcd(m, n) = 1. - Andrew Howroyd, Aug 07 2018

Links

• Antti Karttunen, Table of n, a(n) for n = 1..2101
• 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. A239612, A239613, A239614, A239615, A079458, A053191, A227499, A244342.

Programs

Mathematica

g2[n_] := Sum[If[GCD[x^2 + y^2, n] == 1, GCD[x^2 + y^2 - 1, n], 0], {x, 1, n}, {y, 1, n}]; Array[g2,100]

PARI

a(n) = {s = 0; for (x=1, n, for (y=1, n, if (gcd(x^2+y^2,n) == 1, s += gcd(x^2+y^2-1,n)););); s;} \\ Michel Marcus, Jun 29 2014