A061391 a(n) = t(n,3) = Sum_{d|n} tau(d^3), where tau(n) = number of divisors of n, cf. A000005.

1, 5, 5, 12, 5, 25, 5, 22, 12, 25, 5, 60, 5, 25, 25, 35, 5, 60, 5, 60, 25, 25, 5, 110, 12, 25, 22, 60, 5, 125, 5, 51, 25, 25, 25, 144, 5, 25, 25, 110, 5, 125, 5, 60, 60, 25, 5, 175, 12, 60, 25, 60, 5, 110, 25, 110, 25, 25, 5, 300, 5, 25, 60, 70, 25, 125, 5, 60, 25, 125, 5, 264

Offset

1,2

Comments

Inverse Mobius transform of A048785. - R. J. Mathar, Feb 09 2011

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

t(n, k) = Sum_{d|n} tau(d^k) is multiplicative: if the canonical factorization of n = Product p^e(p) over primes then t(n, k) = Product t(p^e(p), k), t(p^e(p), k) = (1/2) *(k*e(p)+2)*(e(p)+1).

For k=2 we get an interesting identity: Sum_{d|n} tau(d^2)=(tau(n))^2, cf. A048691, A035116.

a(n) = Sum_{d|n} tau(n*d). - Benoit Cloitre, Nov 30 2002

G.f.: Sum_{n>=1} tau(n^3)*x^n/(1-x^n). - Joerg Arndt, Jan 01 2011

Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 + 2/p^s). - Vaclav Kotesovec, May 15 2021

Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Aug 20 2021

Mathematica

f[p_, e_] := (3*e^2 + 5*e + 2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

Prog

(PARI)
A061391 = n -> sumdiv(n, d, numdiv(d^3));
for(n=1, 10000, write("b061391.txt", n, " ", A061391(n)));
\\ Antti Karttunen, Jan 17 2017
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, May 15 2021
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 3*X^2 + 2*X^3)/(1 - X)^5)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

Crossrefs

Cf. t(n, 0) = A000005(n), t(n, 1) = A007425(n), t(n, 2) = A035116(n).

Cf. A048691.

Sequence in context: A098331 A222575 A222681 * A168336 A123133 A328367

Adjacent sequences: A061388 A061389 A061390 * A061392 A061393 A061394

Keyword

nonn,mult

Author

Vladeta Jovovic, Apr 29 2001

Status

approved