A072079 Sum of 3-smooth divisors of n.

1, 3, 4, 7, 1, 12, 1, 15, 13, 3, 1, 28, 1, 3, 4, 31, 1, 39, 1, 7, 4, 3, 1, 60, 1, 3, 40, 7, 1, 12, 1, 63, 4, 3, 1, 91, 1, 3, 4, 15, 1, 12, 1, 7, 13, 3, 1, 124, 1, 3, 4, 7, 1, 120, 1, 15, 4, 3, 1, 28, 1, 3, 13, 127, 1, 12, 1, 7, 4, 3, 1, 195, 1, 3, 4, 7, 1, 12, 1, 31, 121

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = (2^(A007814(n)+1)-1)*(3^(A007949(n)+1)-1)/2.

a(n) = A000203(A065331(n)).

Multiplicative with a(2^e) = 2^(e+1)-1, a(3^e) = (3^(e+1)-1)/2, a(p^e) = 1, p>3. Christian G. Bower, May 20 2005

From Amiram Eldar, Dec 01 2022: (Start)

Dirichlet g.f.: zeta(s)*(2^s/(2^s-2))*(3^s/(3^s-3)).

Sum_{k=1..n} a(k) ~ c_1 * (n * log(n)^2 + c_2 * n * log(n) + c_3 * n), where c_1 = 1/(2*log(2)*log(3)) = 0.656598..., c_2 = (2*gamma - 2 + log(6)) = 0.9461907..., and c_3 = (log(6)^2 + log(2)*log(3))/6 - (log(6)-2)*(1-gamma) - 2*gamma_1 = 0.895656..., gamma is Euler's constant (A001620), and gamma_1 is the 1st Stieltjes constant (A082633). (End)

Mathematica

f[p_, e_] := If[p > 3, 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

Prog

(PARI) a(n) = (2^(valuation(n, 2)+1)-1)*(3^(valuation(n, 3)+1)-1)/2; \\ Amiram Eldar, Dec 01 2022

Crossrefs

Cf. A000203, A003586, A065331, A072078.

Cf. A001620, A082633.

Sequence in context: A093087 A376212 A130893 * A391185 A116073 A382314

Adjacent sequences: A072076 A072077 A072078 * A072080 A072081 A072082

Keyword

nonn,mult

Author

Reinhard Zumkeller, Jun 13 2002

Status

approved