A202994 a(n) = sigma(n^4).

1, 31, 121, 511, 781, 3751, 2801, 8191, 9841, 24211, 16105, 61831, 30941, 86831, 94501, 131071, 88741, 305071, 137561, 399091, 338921, 499255, 292561, 991111, 488281, 959171, 797161, 1431311, 732541, 2929531, 954305, 2097151, 1948705, 2750971, 2187581

Offset: 1

Author

Paul D. Hanna, Dec 27 2011

Keywords

• nonn
• mult

Comments

Here sigma(n^4) denotes the sums of divisors of n^4.

Examples

			L.g.f.: L(x) = x + 31*x^2/2 + 121*x^3/3 + 511*x^4/4 + 781*x^5/5 + 3751*x^6/6 +...
where exp(L(x)) = 1 + x + 16*x^2 + 56*x^3 + 296*x^4 + 1052*x^5 + 4952*x^6 +...+ A202993(n)*x^n +...
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000203 (sigma), A000583, A013663, A065764, A175926, A202993.

Programs

Mathematica

DivisorSigma[1,Range[40]^4] (* Harvey P. Dale, Jan 29 2012 *)
f[p_, e_] := (p^(4*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2020 *)

PARI

{a(n)=sigma(n^4)}

Python

from math import prod
from sympy import factorint
def A202994(n): return prod((p**((e<<2)+1)-1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Oct 25 2023

Formula

a(11*n) == 0 (mod 5) iff gcd(n,11) = 1.

Logarithmic derivative of A202993.

Multiplicative with a(p^e) = (p^(4*e+1)-1)/(p-1) for prime p. - Andrew Howroyd, Jul 23 2018

a(n) = A000203(A000583(n)). - Michel Marcus, Sep 10 2020

Sum_{k>=1} 1/a(k) = 1.04483665108279017775482622699860068916340892303889072390102812885655694752... - Vaclav Kotesovec, Sep 20 2020

Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)/5) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.3840585791... . - Amiram Eldar, Nov 05 2022

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018