A239442 a(n) = phi(n^7).

1, 64, 1458, 8192, 62500, 93312, 705894, 1048576, 3188646, 4000000, 17715610, 11943936, 57921708, 45177216, 91125000, 134217728, 386201104, 204073344, 846825858, 512000000, 1029193452, 1133799040, 3256789558, 1528823808, 4882812500, 3706989312, 6973568802, 5782683648, 16655052988

Offset

1,2

Comments

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

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

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

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

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

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

Maple

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

Mathematica

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

Prog

(PARI) a(n) = n^6*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.

Phi_2(n) = A079458.

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

Phi_4(n) = A227499.

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

Phi_6(n) = A238534.

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

Phi_8(n) = A239441.

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

Sequence in context: A333812 A283812 A264086 * A240930 A208313 A145218

Adjacent sequences: A239439 A239440 A239441 * A239443 A239444 A239445

Keyword

nonn,mult

Author

José María Grau Ribas, Mar 19 2014

Status

approved