

A048785


a(0) = 0; a(n) = tau(n^3), where tau = number of divisors (A000005).


14



0, 1, 4, 4, 7, 4, 16, 4, 10, 7, 16, 4, 28, 4, 16, 16, 13, 4, 28, 4, 28, 16, 16, 4, 40, 7, 16, 10, 28, 4, 64, 4, 16, 16, 16, 16, 49, 4, 16, 16, 40, 4, 64, 4, 28, 28, 16, 4, 52, 7, 28, 16, 28, 4, 40, 16, 40, 16, 16, 4, 112, 4, 16, 28, 19, 16, 64, 4, 28, 16
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OFFSET

0,3


COMMENTS

The inverse Mobius transform of A074816.  R. J. Mathar, Feb 09 2011
a(n) is also the number of ordered triples (i,j,k) of positive integers such that in, jn, kn and i,j,k are pairwise relatively prime.  Geoffrey Critzer, Jan 11 2015


LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10000


FORMULA

a(n) = Sum_{dn} 3^omega(d), where omega(x) is the number of distinct prime factors in the factorization of x.  Benoit Cloitre, Apr 14 2002
Multiplicative with a(p^e) = 3e+1.  Mitch Harris, Jun 09 2005
L.g.f.: log(Product_{k>=1} (1  x^k)^(3^omega(k)/k)) = Sum_{n>=1} a(n)*x^n/n.  Ilya Gutkovskiy, May 26 2018
For n>0, a(n) = Sum_{dn} mu(d)^2*tau(d)*tau(n/d).  Ridouane Oudra, Nov 18 2019


EXAMPLE

a(6) = 16 because there are 16 divisors of 6^3 = 216: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216.
Also there are 16 ordered triples of divisors of 6 that are pairwise relatively prime: (1,1,1), (1,1,2), (1,1,3), (1,1,6), (1, 2, 1), (1,2,3), (1,3,1), (1,3,2), (1,6,1), (2,1,1), (2,1,3), (2,3,1), (3,1,1), (3,1,2),(3,2,1), (6,1,1).


MAPLE

seq(numtheory:tau(n^3), n=0..100); # Robert Israel, Jan 11 2015


MATHEMATICA

Join[{1}, Table[Product[3 k + 1, {k, FactorInteger[n][[All, 2]]}], {n, 2, 69}]] (* Geoffrey Critzer, Jan 11 2015 *)
Join[{0}, DivisorSigma[0, Range[70]^3]] (* Harvey P. Dale, Jan 23 2016 *)


PROG

(PARI) A048785(n) = if(!n, n, numdiv(n^3)); \\ Antti Karttunen, May 19 2017


CROSSREFS

Cf. A000005, A001221, A048691, A074816, A144943.
Sequence in context: A331619 A258972 A146564 * A271781 A243454 A204008
Adjacent sequences: A048782 A048783 A048784 * A048786 A048787 A048788


KEYWORD

nonn,mult


AUTHOR

N. J. A. Sloane


STATUS

approved



