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 A048785 a(0) = 0; a(n) = tau(n^3), where tau = number of divisors (A000005). 25
 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: A374104 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 3 05:44 EDT 2024. Contains 374875 sequences. (Running on oeis4.)