OFFSET
1,2
COMMENTS
In general, for m > 2, Sum_{k=1..n} phi(k) * floor(n/k)^m ~ zeta(m-1) * n^m / zeta(m). - Vaclav Kotesovec, May 23 2021
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^5.
G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^5.
a(n) ~ Pi^4 * n^5 / (90*zeta(5)). - Vaclav Kotesovec, May 23 2021
MATHEMATICA
a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)
PROG
(PARI) a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, sum(l=1, n, sum(m=1, n, gcd([i, j, k, l, m]))))));
(PARI) a(n) = sum(k=1, n, eulerphi(k)*(n\k)^5);
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+26*x^k+66*x^(2*k)+26*x^(3*k)+x^(4*k))/(1-x^k)^5)/(1-x))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 22 2021
STATUS
approved