A361147 a(n) = sigma(n)^3.

1, 27, 64, 343, 216, 1728, 512, 3375, 2197, 5832, 1728, 21952, 2744, 13824, 13824, 29791, 5832, 59319, 8000, 74088, 32768, 46656, 13824, 216000, 29791, 74088, 64000, 175616, 27000, 373248, 32768, 250047, 110592, 157464, 110592, 753571, 54872, 216000, 175616

Offset: 1

Author

Vaclav Kotesovec, Mar 02 2023

Keywords

• nonn
• mult

Crossrefs

Cf. A000203, A024916, A072861, A072379, A361179.

Cf. A000005, A000578, A035116, A319089.

Programs

Mathematica

Table[DivisorSigma[1, n]^3, {n, 1, 50}]

PARI

a(n) = sigma(n)^3;

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X*(2 + 2*p + p^2*X)) / ((1-X)*(1-p*X)*(1-p^2*X)*(1-p^3*X)))[n], ", "))

Formula

Multiplicative with a(p^e) = ((p^(e+1)-1)/(p-1))^3.

Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) * Product_{primes p} (1 + 1/p^(2*s-3) + 2/p^(s-1) + 2/p^(s-2)).

Sum_{k=1..n} a(k) ~ c * Pi^6 * zeta(3) * n^4 / 2160, where c = Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 2.83598357433419286770442457158038489640898183...

a(n) = A000578(A000203(n)).