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

Author

Reinhard Zumkeller, Jun 13 2002

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A000203, A003586, A065331, A072078.

Cf. A001620, A082633.

Programs

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 *)

PARI

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

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)