OFFSET
1,2
COMMENTS
In general, if m > 1 and a(n) = Sum_{d|n} phi(n/d) * binomial(d + m - 1, m) then Sum_{k=1..n} a(k) ~ zeta(m) * n^(m+1) / ((m+1)! * zeta(m+1)). - Vaclav Kotesovec, May 23 2021
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{d|n} phi(n/d) * binomial(d+6, 7).
G.f.: Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^8.
Sum_{k=1..n} a(k) ~ 15*zeta(7)*n^8 / (64*Pi^8). - Vaclav Kotesovec, May 23 2021
MATHEMATICA
a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + 6, 7] &]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
PROG
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+6, 7));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^8))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Apr 17 2021
STATUS
approved