OFFSET
1,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{d|n} phi(n/d) * tau(d^4).
a(n) = n * Sum_{d|n} 4^omega(d) / d.
If p is prime, a(p) = 4 + p.
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = (p^e*(p + 3) - 4)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 3/p^2) = 2.4997873122... . (End)
MATHEMATICA
Table[Sum[DivisorSigma[0, GCD[k, n]^4], {k, n}], {n, 100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
f[p_, e_] := (p^e*(p + 3) - 4)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)
PROG
(PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n)^4));
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^4));
(PARI) a(n) = n*sumdiv(n, d, 4^omega(d)/d);
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 12 2021
STATUS
approved