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A048785 a(0) = 0; a(n) = tau(n^3), where tau = number of divisors (A000005). 20
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 (list; graph; refs; listen; history; text; internal format)
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 i|n, j|n, k|n 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_{d|n} 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_{d|n} mu(d)^2*tau(d)*tau(n/d). - Ridouane Oudra, Nov 18 2019

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

Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Aug 20 2021

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[{0, 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

(PARI) print1("0, "); for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, May 15 2021

print1("0, "); for(n=1, 100, print1(direuler(p=2, n, (1 - 3*X^2 + 2*X^3)/(1 - X)^4)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

(Python)

from math import prod

from sympy import factorint

def A048785(n): return 0 if n == 0 else prod(3*e+1 for e in factorint(n).values()) # Chai Wah Wu, May 10 2022

(Python) from sympy import divisor_count

def A048785(n): return divisor_count(n**3) # Karl-Heinz Hofmann, May 10 2022

CROSSREFS

Cf. A000005, A001221, A048691, A074816, A144943, A353551.

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

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Last modified August 16 03:06 EDT 2022. Contains 356150 sequences. (Running on oeis4.)