A239442 - OEIS

%I #40 Dec 06 2020 02:47:39

%S 1,64,1458,8192,62500,93312,705894,1048576,3188646,4000000,17715610,

%T 11943936,57921708,45177216,91125000,134217728,386201104,204073344,

%U 846825858,512000000,1029193452,1133799040,3256789558,1528823808,4882812500,3706989312,6973568802,5782683648,16655052988

%N a(n) = phi(n^7).

%C Number of solutions of the equation gcd(x_1^2 + ... + x_7^2, n)=1 with 0 < x_i <= n.

%H Seiichi Manyama, <a href="/A239442/b239442.txt">Table of n, a(n) for n = 1..10000</a>

%H C. Calderón, J. M. Grau, A. Oller-Marcén, and László Tóth, <a href="http://arxiv.org/abs/1403.7878">Counting invertible sums of squares modulo n and a new generalization of Euler totient function</a>, arXiv:1403.7878 [math.NT], 2014.

%F a(n) = n^6*phi(n).

%F Dirichlet g.f.: zeta(s - 7) / zeta(s - 6). The n-th term of the Dirichlet inverse is n^6 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega=A001221. - _Álvar Ibeas_, Nov 24 2017

%F Sum_{k=1..n} a(k) ~ 3*n^8 / (4*Pi^2). - _Vaclav Kotesovec_, Feb 02 2019

%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^8 - p^7 - p + 1)) = 1.01646280485545934937... - _Amiram Eldar_, Dec 06 2020

%p with(numtheory); A239442:=n->phi(n^7); seq(A239442(n), n=1..100); # _Wesley Ivan Hurt_, Apr 01 2014

%t Table[EulerPhi[n^7], {n, 100}]

%o (PARI) a(n) = n^6*eulerphi(n); \\ _Michel Marcus_, Mar 10 2018

%Y Defining Phi_k(n):= number of solutions of the equation gcd(x_1^2 + ... + x_k^2, n) = 1 with 0 < x_i <= n.

%Y Phi_1(n) = phi(n) = A000010.

%Y Phi_2(n) = A079458.

%Y Phi_3(n) = phi(n^3) = n^2*phi(n)= A053191.

%Y Phi_4(n) = A227499.

%Y Phi_5(n) = phi(n^5) = n^4*phi(n)= A238533.

%Y Phi_6(n) = A238534.

%Y Phi_7(n) = phi(n^7) = n^6*phi(n)= A239442.

%Y Phi_8(n) = A239441.

%Y Phi_9(n) = phi(n^9) = n^8*phi(n)= A239443.

%K nonn,mult

%O 1,2

%A _José María Grau Ribas_, Mar 19 2014