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A061391 a(n) = t(n,3) = Sum_{d|n} tau(d^3), where tau(n) = number of divisors of n, cf. A000005. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 17 01:53 EDT 2021. Contains 347478 sequences. (Running on oeis4.)