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A072078 Number of 3-smooth divisors of n.

%I A072078 #42 Sep 08 2022 08:45:06
%S A072078 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,
%T A072078 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,
%U A072078 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
%N A072078 Number of 3-smooth divisors of n.
%H A072078 Antti Karttunen, <a href="/A072078/b072078.txt">Table of n, a(n) for n = 1..19683</a>
%F A072078 a(n) = A000005(A065331(n)).
%F A072078 a(n) = (A007814(n) + 1)*(A007949(n) + 1).
%F A072078 1/Product_{k>0} (1 - x^k + x^(2*k))^a(k) is g.f. for A000041(). - _Vladeta Jovovic_, Jun 07 2004
%F A072078 From _Christian G. Bower_, May 20 2005: (Start)
%F A072078 Multiplicative with a(2^e) = a(3^e) = e+1, a(p^e) = 1, p>3.
%F A072078 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)
%F A072078 a(n) = Sum_{d divides n} mu(6d)*tau(n/d). - _Benoit Cloitre_, Jun 21 2007
%F A072078 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
%F A072078 Sum_{k=1..n} a(k) ~ 3*n. - _Vaclav Kotesovec_, Nov 20 2021
%t A072078 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 *)
%o A072078 (PARI) a(n)=sumdiv(n,d,moebius(6*d)*numdiv(n/d)) \\ _Benoit Cloitre_, Jun 21 2007
%o A072078 (PARI) A072078(n) = ((1+valuation(n,2))*(1+valuation(n,3))); \\ _Antti Karttunen_, Dec 03 2018
%o A072078 (Magma) [(Valuation(n,2)+1)*(Valuation(n,3)+1): n in [1..120]]; // _Vincenzo Librandi_, Mar 24 2015
%Y A072078 Cf. A000005, A003586, A007814, A007949, A072079, A122841, A322026.
%K A072078 nonn,mult
%O A072078 1,2
%A A072078 _Reinhard Zumkeller_, Jun 13 2002
%E A072078 More terms from _Benoit Cloitre_, Jun 21 2007