%I #67 May 10 2022 13:57:26
%S 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,
%T 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,
%U 28,4,40,16,40,16,16,4,112,4,16,28,19,16,64,4,28,16
%N a(0) = 0; a(n) = tau(n^3), where tau = number of divisors (A000005).
%C The inverse Mobius transform of A074816. - _R. J. Mathar_, Feb 09 2011
%C 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
%H Antti Karttunen, <a href="/A048785/b048785.txt">Table of n, a(n) for n = 0..10000</a>
%F 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
%F Multiplicative with a(p^e) = 3e+1. - _Mitch Harris_, Jun 09 2005
%F 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
%F For n>0, a(n) = Sum_{d|n} mu(d)^2*tau(d)*tau(n/d). - _Ridouane Oudra_, Nov 18 2019
%F Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 2/p^s). - _Vaclav Kotesovec_, May 15 2021
%F Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - _Vaclav Kotesovec_, Aug 20 2021
%e 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.
%e 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).
%p seq(numtheory:-tau(n^3), n=0..100); # _Robert Israel_, Jan 11 2015
%t Join[{0,1},Table[Product[3 k + 1, {k, FactorInteger[n][[All, 2]]}], {n, 2, 69}]] (* _Geoffrey Critzer_, Jan 11 2015 *)
%t Join[{0},DivisorSigma[0,Range[70]^3]] (* _Harvey P. Dale_, Jan 23 2016 *)
%o (PARI) A048785(n) = if(!n,n,numdiv(n^3)); \\ _Antti Karttunen_, May 19 2017
%o (PARI) print1("0, "); for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, May 15 2021
%o 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
%o (Python)
%o from math import prod
%o from sympy import factorint
%o 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
%o (Python) from sympy import divisor_count
%o def A048785(n): return divisor_count(n**3) # _Karl-Heinz Hofmann_, May 10 2022
%Y Cf. A000005, A001221, A048691, A074816, A144943, A353551.
%K nonn,mult
%O 0,3
%A _N. J. A. Sloane_