A238533 Number of solutions to gcd(x^2 + y^2 + z^2 + t^2 + h^2, n) = 1 with x,y,z,t,h in [0,n-1].

1, 16, 162, 512, 2500, 2592, 14406, 16384, 39366, 40000, 146410, 82944, 342732, 230496, 405000, 524288, 1336336, 629856, 2345778, 1280000, 2333772, 2342560, 6156502, 2654208, 7812500, 5483712, 9565938, 7375872, 19803868, 6480000, 27705630, 16777216, 23718420

Offset: 1

Author

José María Grau Ribas, Feb 28 2014

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Param Parekh, Paavan Parekh, Sourav Deb, and Manish K. Gupta, On the Classification of Weierstrass Elliptic Curves over Z_n, arXiv:2310.11768 [cs.CR], 2023. See p. 9.

Crossrefs

Cf. A079458, A227499, A238534.

Cf. n^k * a(n): A000010 (k=-4), A002618 (k=-3), A053191 (k=-2), A189393 (k=-1), A239442 (k=2), A239443 (k=4).

Programs

Mathematica

g[n_, 5] := g[n, 5] = Sum[If[GCD[x^2 + y^2 + z^2 + t^2 + h^2, n] == 1, 1, 0], {x, n}, {y, n}, {z, n}, {t, n}, {h, n}];Table[g[n,5] , {n, 1, 15}]
Table[n^4 * EulerPhi[n], {n, 1, 33}] (* Amiram Eldar, Dec 06 2020 *)

Formula

From Álvar Ibeas, Nov 24 2017: (Start)

a(n) = phi(n^5) = n^4 * phi(n), where phi=A000010.

Dirichlet g.f.: zeta(s - 5) / zeta(s - 4). The n-th term of the Dirichlet inverse is n^4 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega = A001221.

(End)

Sum_{k=1..n} a(k) ~ n^6 / Pi^2. - Vaclav Kotesovec, Feb 02 2019

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^6 - p^5 - p + 1)) = 1.07162935672651489627... - Amiram Eldar, Dec 06 2020