A239443 a(n) = phi(n^9), where phi = A000010.

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

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