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
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).
KEYWORD
nonn,mult
AUTHOR
José María Grau Ribas, Mar 22 2014
STATUS
approved