OFFSET
1,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_5(d).
a(n) = Sum_{d|n} d^(5-m) * phi(d^m) for m > 0.
G.f.: Sum_{k>=1} k^(5-m) * phi(k^m) * x^k/(1 - x^k) for m > 0.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-4).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(2) = 2*Pi^4/315 = 0.6184704192... (1/A157292). (End)
MATHEMATICA
f[p_, e_] := (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)
PROG
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 5));
(PARI) a(n) = sumdiv(n, d, eulerphi(d^5));
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 18 2024
STATUS
approved