A077457 a(n) = sigma_4(n^4)/sigma_2(n^4).

1, 205, 5905, 52429, 375601, 1210525, 5649505, 13421773, 38742049, 76998205, 212601841, 309593245, 810932305, 1158148525, 2217923905, 3435973837, 6951703105, 7942120045, 16936647121, 19692384829, 33360327025, 43583377405, 78163228705, 79255569565, 146719125601

Offset: 1

Author

Benoit Cloitre, Nov 30 2002

Keywords

• nonn
• easy
• mult

Comments

sigma_y(n^x) divides sigma_x(n^x) for all n if y divides x.

Links

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

Crossrefs

Cf. A001157, A001159, A013667, A057660, A077454, A077455, A077456.

Programs

Mathematica

f[p_, e_] := (p^(8*e+2) + 1)/(p^2 + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 25] (* Amiram Eldar, Sep 09 2020 *)

PARI

a(n)=sumdiv(n^4,d,d^4)/sumdiv(n^4,d,d^2)

PARI

a(n) = my(f=factor(n^4)); sigma(f, 4)/sigma(f, 2); \\ Michel Marcus, Sep 09 2020

Formula

a(n) = A001159(n^4)/A001157(n^4).

Multiplicative with a(p^e) = (p^(8*e+2) + 1)/(p^2 + 1). - Amiram Eldar, Sep 09 2020

Sum_{k=1..n} a(k) ~ c * n^9, where c = (zeta(9)/9) * Product_{p prime} (1 - 1/p^3 + 1/p^5 - 1/p^7) = 0.09549806119... . - Amiram Eldar, Oct 28 2022