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%I #43 Oct 25 2023 09:30:52
%S 1,16,162,512,2500,2592,14406,16384,39366,40000,146410,82944,342732,
%T 230496,405000,524288,1336336,629856,2345778,1280000,2333772,2342560,
%U 6156502,2654208,7812500,5483712,9565938,7375872,19803868,6480000,27705630,16777216,23718420
%N 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].
%H Amiram Eldar, <a href="/A238533/b238533.txt">Table of n, a(n) for n = 1..10000</a>
%H Param Parekh, Paavan Parekh, Sourav Deb, and Manish K. Gupta, <a href="https://arxiv.org/abs/2310.11768">On the Classification of Weierstrass Elliptic Curves over Z_n</a>, arXiv:2310.11768 [cs.CR], 2023. See p. 9.
%F From _Álvar Ibeas_, Nov 24 2017: (Start)
%F a(n) = phi(n^5) = n^4 * phi(n), where phi=A000010.
%F 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.
%F (End)
%F Sum_{k=1..n} a(k) ~ n^6 / Pi^2. - _Vaclav Kotesovec_, Feb 02 2019
%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^6 - p^5 - p + 1)) = 1.07162935672651489627... - _Amiram Eldar_, Dec 06 2020
%t 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}]
%t Table[n^4 * EulerPhi[n], {n, 1, 33}] (* _Amiram Eldar_, Dec 06 2020 *)
%Y Cf. A079458, A227499, A238534.
%Y Cf. n^k * a(n): A000010 (k=-4), A002618 (k=-3), A053191 (k=-2), A189393 (k=-1), A239442 (k=2), A239443 (k=4).
%K nonn,mult
%O 1,2
%A _José María Grau Ribas_, Feb 28 2014
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