A372965 a(n) = Sum_{k = 1..n} ( n/gcd(k, n) )^4.

1, 17, 163, 529, 2501, 2771, 14407, 16913, 39529, 42517, 146411, 86227, 342733, 244919, 407663, 541201, 1336337, 671993, 2345779, 1323029, 2348341, 2488987, 6156503, 2756819, 7815001, 5826461, 9605467, 7621303, 19803869, 6930271, 27705631, 17318417, 23864993

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

Cf. A057660, A068963, A068970.

Cf. A001160, A008683.

Cf. A372966.

Cf. A013661, A013664, A157292.

Sequence in context: A198859 A160295 A125645 * A068518 A155664 A096192

Adjacent sequences: A372962 A372963 A372964 * A372966 A372967 A372968

Keyword

nonn,mult

Author

Seiichi Manyama, May 18 2024

Status

approved