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

1,2

Links

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

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)

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 *)

Prog

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

Crossrefs

Row sums of A129353.

Cf. A000005, A001511, A001620, A129628.

Cf. A007814, A099777, A115364, A001227.

Sequence in context: A021758 A040008 A129354 * A245691 A071044 A077066

Adjacent sequences: A129625 A129626 A129627 * A129629 A129630 A129631

Keyword

nonn,mult,easy

Author

Ralf Stephan, May 31 2007

Status

approved