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A072078 Number of 3-smooth divisors of n. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Multiplicative with a(2^e) = a(3^e) = e+1, a(p^e) = 1, p>3. Dirichlet g.f. is 2/((1-1/2^s)*(1-1/3^s)) * prod{n is a prime > 3}(1/(1-1/n^s)). - Christian G. Bower, May 20 2005

LINKS

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

FORMULA

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

a(n) = sum_{d divides n} mu(6d)*tau(n/d). - Benoit Cloitre, Jun 21 2007

Dirichlet g.f.: zeta(s)/((1-2^s)*(1-3^s)). - Ralf Stephan, Mar 24 2015

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: A071450 A175457 A322025 * 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

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Last modified February 28 11:08 EST 2020. Contains 332323 sequences. (Running on oeis4.)