A072078 Number of 3-smooth divisors of n.
1, 2, 2, 3, 1, 4, 1, 4, 3, 2, 1, 6, 1, 2, 2, 5, 1, 6, 1, 3, 2, 2, 1, 8, 1, 2, 4, 3, 1, 4, 1, 6, 2, 2, 1, 9, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 10, 1, 2, 2, 3, 1, 8, 1, 4, 2, 2, 1, 6, 1, 2, 3, 7, 1, 4, 1, 3, 2, 2, 1, 12, 1, 2, 2, 3, 1, 4, 1, 5, 5, 2, 1, 6, 1, 2, 2, 4, 1, 6, 1, 3, 2, 2, 1
Offset: 1
Links
- Antti Karttunen, Table of n, a(n) for n = 1..19683
Programs
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Magma
[(Valuation(n,2)+1)*(Valuation(n,3)+1): n in [1..120]]; // Vincenzo Librandi, Mar 24 2015
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Mathematica
a[n_] := DivisorSum[n, MoebiusMu[6*#]*DivisorSigma[0, n/#] &]; Array[a, 100] (* or *) a[n_] := ((1+IntegerExponent[n, 2])*(1+IntegerExponent[n, 3])); Array[a, 100] (* Amiram Eldar, Dec 03 2018 from the pari codes *)
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PARI
a(n)=sumdiv(n,d,moebius(6*d)*numdiv(n/d)) \\ Benoit Cloitre, Jun 21 2007
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PARI
A072078(n) = ((1+valuation(n,2))*(1+valuation(n,3))); \\ Antti Karttunen, Dec 03 2018
Formula
1/Product_{k>0} (1 - x^k + x^(2*k))^a(k) is g.f. for A000041(). - Vladeta Jovovic, Jun 07 2004
From Christian G. Bower, May 20 2005: (Start)
Multiplicative with a(2^e) = a(3^e) = e+1, a(p^e) = 1, p>3.
Dirichlet g.f.: 1/((1-1/2^s)*(1-1/3^s))^2 * Product{p prime > 3}(1/(1-1/p^s)). [corrected by Vaclav Kotesovec, Nov 20 2021] (End)
a(n) = Sum_{d divides n} mu(6d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
Dirichlet g.f.: zeta(s)/((1-1/2^s)*(1-1/3^s)). - Ralf Stephan, Mar 24 2015; corrected by Vaclav Kotesovec, Nov 20 2021
Sum_{k=1..n} a(k) ~ 3*n. - Vaclav Kotesovec, Nov 20 2021
Extensions
More terms from Benoit Cloitre, Jun 21 2007