A077454 a(n) = sigma_3(n^3)/sigma(n^3).

1, 39, 511, 2359, 12621, 19929, 101179, 149943, 368089, 492219, 1611831, 1205449, 4457701, 3945981, 6449331, 9588151, 22722609, 14355471, 44576623, 29772939, 51702469, 62861409, 141611691, 76620873, 196890121, 173850339, 268218727, 238681261, 574336533, 251523909

Offset: 1

Author

Benoit Cloitre, Nov 30 2002

Keywords

• nonn
• easy
• mult

Examples

			a(2) = sigma_3(2^3)/sigma(2^3) = 585/15 = 39.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000578, A001158, A013665, A057660, A077455, A077456.

Programs

Mathematica

f[p_, e_] := (p^(6*e+2) + p^(3*e+1) + 1)/(p^2 + p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 30] (* Amiram Eldar, Sep 09 2020 *)

PARI

a(n)=sumdiv(n^3,d,d^3)/sigma(n^3)

PARI

a(n) = my(f=factor(n^3)); sigma(f, 3)/sigma(f); \\ Michel Marcus, Sep 09 2020

Formula

a(n) = A001158(n^3)/A000203(n^3).

Multiplicative with a(p^e) = (p^(6*e+2) + p^(3*e+1) + 1)/(p^2 + p + 1). - Amiram Eldar, Sep 09 2020

Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)*Pi^4/630) * Product_{p prime} (1 - 1/p^2 - 1/p^6 + 1/p^7 - 1/p^8 + 1/p^9) = 0.09343400455... . - Amiram Eldar, Oct 28 2022