A319089 a(n) = tau(n)^3, where tau is A000005.

1, 8, 8, 27, 8, 64, 8, 64, 27, 64, 8, 216, 8, 64, 64, 125, 8, 216, 8, 216, 64, 64, 8, 512, 27, 64, 64, 216, 8, 512, 8, 216, 64, 64, 64, 729, 8, 64, 64, 512, 8, 512, 8, 216, 216, 64, 8, 1000, 27, 216, 64, 216, 8, 512, 64, 512, 64, 64, 8, 1728, 8, 64, 216, 343

Offset

1,2

Links

Table of n, a(n) for n=1..64.

Eric Weisstein's World of Mathematics, Divisor Function.

Formula

Multiplicative with a(p^e) = (e+1)^3. - Amiram Eldar, Dec 31 2022

a(n) = Sum_{d1|n} Sum_{d2|n} tau(d1*d2). - Ridouane Oudra, Mar 07 2023

From Vaclav Kotesovec, Mar 09 2023: (Start)

Dirichlet g.f.: Product_{p prime} p^(2*s) * (1 + 4*p^s + p^(2*s)) / (p^s - 1)^4.

Dirichlet g.f.: zeta(s)^8 * Product_{p prime} (1 - 9/p^(2*s) + 16/p^(3*s) - 9/p^(4*s) + 1/p^(6*s)), (with a product that converges for s=1). (End)

Maple

with(numtheory): seq(tau(n)^3, n=1..100); # Ridouane Oudra, Mar 07 2023

Mathematica

DivisorSigma[0, Range[100]]^3

Prog

(PARI) a(n) = numdiv(n)^3; \\ Altug Alkan, Sep 10 2018
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X + X^2)/(1 - X)^4)[n], ", ")) \\ Vaclav Kotesovec, Mar 09 2023

Crossrefs

Cf. A000005, A006218, A035116, A061502, A318755 (partial sums).

Sequence in context: A037018 A246310 A318542 * A003873 A077110 A333625

Adjacent sequences: A319086 A319087 A319088 * A319090 A319091 A319092

Keyword

nonn,easy,mult

Author

Vaclav Kotesovec, Sep 10 2018

Status

approved