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

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..19683

Formula

a(n) = A000005(A065331(n)).

a(n) = (A007814(n) + 1)*(A007949(n) + 1).

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

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 *)

Prog

(PARI) a(n)=sumdiv(n, d, moebius(6*d)*numdiv(n/d)) \\ Benoit Cloitre, Jun 21 2007
(PARI) A072078(n) = ((1+valuation(n, 2))*(1+valuation(n, 3))); \\ Antti Karttunen, Dec 03 2018
(Magma) [(Valuation(n, 2)+1)*(Valuation(n, 3)+1): n in [1..120]]; // Vincenzo Librandi, Mar 24 2015

Crossrefs

Cf. A000005, A003586, A007814, A007949, A072079, A122841, A322026.

Sequence in context: A175457 A322025 A339526 * A322316 A260439 A182471

Adjacent sequences: A072075 A072076 A072077 * A072079 A072080 A072081

Keyword

nonn,mult

Author

Reinhard Zumkeller, Jun 13 2002

Extensions

More terms from Benoit Cloitre, Jun 21 2007

Status

approved