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
KEYWORD
nonn,mult
AUTHOR
STATUS
approved