A129628 Inverse Moebius transform of A001511.

1, 3, 2, 6, 2, 6, 2, 10, 3, 6, 2, 12, 2, 6, 4, 15, 2, 9, 2, 12, 4, 6, 2, 20, 3, 6, 4, 12, 2, 12, 2, 21, 4, 6, 4, 18, 2, 6, 4, 20, 2, 12, 2, 12, 6, 6, 2, 30, 3, 9, 4, 12, 2, 12, 4, 20, 4, 6, 2, 24, 2, 6, 6, 28, 4, 12, 2, 12, 4, 12, 2, 30, 2, 6, 6, 12, 4, 12, 2, 30, 5, 6, 2, 24, 4, 6, 4, 20

Offset: 1

Author

Ralf Stephan, May 31 2007

Keywords

• nonn
• mult
• easy

Comments

Dirichlet convolution of A000005 with A209229. - Ridouane Oudra, Jul 25 2025

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

Crossrefs

Row sums of A129353.

Cf. A000005, A001511, A001620, A129628.

Cf. A007814, A099777, A115364, A001227, A209229.

Programs

Maple

seq(add(padic[ordp](2*d, 2), d in numtheory[divisors](n)), n=1..100); # Ridouane Oudra, Sep 30 2024

Mathematica

f[p_, e_] := If[p==2, (e+1)*(e+2)/2, e+1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 30 2020 *)

PARI

a(n)={sumdiv(n, d, 1 + valuation(d, 2))} \\ Andrew Howroyd, Aug 04 2018

Formula

a(2n) = a(n) + A000005(2n), a(2n+1) = A000005(2n+1).

Dirichlet g.f.: zeta(s)^2 * 2^s/(2^s-1). - Ralf Stephan, Jun 17 2007, corrected by Vaclav Kotesovec, Feb 02 2019

a(n) = Sum_{d|n} A001511(d). - Andrew Howroyd, Aug 04 2018

Sum_{k=1..n} a(k) ~ 2*n * (2*gamma - 1 + log(n/2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019

Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = e+1 for p > 2. - Amiram Eldar, Sep 30 2020

From Ridouane Oudra, Sep 30 2024: (Start)

a(n) = Sum_{i=0..A007814(n)} tau(n/2^i).

a(n) = Sum_{d|2*n} A007814(d).

a(n) = (1/2)*A001511(n)*A099777(n).

a(n) = (1/2)*(A001511(n) + 1)*A000005(n).

a(n) = A115364(n)*A001227(n). (End)