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%I #46 Dec 06 2020 02:50:10
%S 1,256,13122,131072,1562500,3359232,34588806,67108864,258280326,
%T 400000000,2143588810,1719926784,9788768652,8854734336,20503125000,
%U 34359738368,111612119056,66119763456,305704134738,204800000000,453874312332,548758735360,1722841676182,880602513408,3051757812500
%N a(n) = phi(n^9), where phi = A000010.
%C Number of solutions of the equation GCD(x_1^2 + ... + x_9^2,n)=1 with 0 < x_i <= n.
%C In general, for m>0, Sum_{k=1..n} phi(k^m) ~ 6 * n^(m+1) / ((m+1)*Pi^2). - _Vaclav Kotesovec_, Feb 02 2019
%H Seiichi Manyama, <a href="/A239443/b239443.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 Dirichlet g.f.: zeta(s - 9) / zeta(s - 8). The n-th term of the Dirichlet inverse is n^8 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega = A001221. - _Álvar Ibeas_, Nov 24 2017
%F a(n) = n^8 * phi(n). - _Altug Alkan_, Mar 10 2018
%F Sum_{k=1..n} a(k) ~ 3*n^10 / (5*Pi^2). - _Vaclav Kotesovec_, Feb 02 2019
%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^10 - p^9 - p + 1)) = 1.00399107654133714629... - _Amiram Eldar_, Dec 06 2020
%p with(numtheory); A239443:=n->phi(n^9); seq(A239443(n), n=1..100); # _Wesley Ivan Hurt_, Apr 01 2014
%t Table[EulerPhi[n^9], {n, 100}]
%o (PARI) a(n) = n^8*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(n).
%Y Phi_2(n) = A079458(n).
%Y Phi_3(n) = phi(n^3) = n^2*phi(n)= A053191(n).
%Y Phi_4(n) = A227499(n).
%Y Phi_5(n) = phi(n^5) = n^4*phi(n)= A238533(n).
%Y Phi_6(n) = A238534(n).
%Y Phi_7(n) = phi(n^7) = n^6*phi(n)= A239442(n).
%Y Phi_8(n) = A239441(n).
%Y Phi_9(n) = phi(n^9) = n^8*phi(n)= A239443(n).
%K nonn,mult
%O 1,2
%A _José María Grau Ribas_, Mar 22 2014