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 A239443 a(n) = phi(n^9), where phi = A000010. 9
 1, 256, 13122, 131072, 1562500, 3359232, 34588806, 67108864, 258280326, 400000000, 2143588810, 1719926784, 9788768652, 8854734336, 20503125000, 34359738368, 111612119056, 66119763456, 305704134738, 204800000000, 453874312332, 548758735360, 1722841676182, 880602513408, 3051757812500 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of solutions of the equation GCD(x_1^2 + ... + x_9^2,n)=1 with 0 < x_i <= n. 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 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 C. Calderón, J. M. Grau, A. Oller-Marcén, and László Tóth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014. FORMULA 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 a(n) = n^8 * phi(n). - Altug Alkan, Mar 10 2018 Sum_{k=1..n} a(k) ~ 3*n^10 / (5*Pi^2). - Vaclav Kotesovec, Feb 02 2019 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^10 - p^9 - p + 1)) = 1.00399107654133714629... - Amiram Eldar, Dec 06 2020 MAPLE with(numtheory); A239443:=n->phi(n^9); seq(A239443(n), n=1..100); # Wesley Ivan Hurt, Apr 01 2014 MATHEMATICA Table[EulerPhi[n^9], {n, 100}] PROG (PARI) a(n) = n^8*eulerphi(n); \\ Michel Marcus, Mar 10 2018 CROSSREFS Defining Phi_k(n):= number of solutions of the equation GCD(x_1^2 + ... + x_k^2,n)=1 with 0 < x_i <= n. Phi_1(n) = phi(n) = A000010(n). Phi_2(n) = A079458(n). Phi_3(n) = phi(n^3) = n^2*phi(n)= A053191(n). Phi_4(n) = A227499(n). Phi_5(n) = phi(n^5) = n^4*phi(n)= A238533(n). Phi_6(n) = A238534(n). Phi_7(n) = phi(n^7) = n^6*phi(n)= A239442(n). Phi_8(n) = A239441(n). Phi_9(n) = phi(n^9) = n^8*phi(n)= A239443(n). Sequence in context: A205652 A284032 A205089 * A240932 A238073 A017032 Adjacent sequences: A239440 A239441 A239442 * A239444 A239445 A239446 KEYWORD nonn,mult AUTHOR José María Grau Ribas, Mar 22 2014 STATUS approved

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Last modified February 24 10:53 EST 2024. Contains 370295 sequences. (Running on oeis4.)