A174238 Inverse Moebius transform of even part of n (A006519).

1, 3, 2, 7, 2, 6, 2, 15, 3, 6, 2, 14, 2, 6, 4, 31, 2, 9, 2, 14, 4, 6, 2, 30, 3, 6, 4, 14, 2, 12, 2, 63, 4, 6, 4, 21, 2, 6, 4, 30, 2, 12, 2, 14, 6, 6, 2, 62, 3, 9, 4, 14, 2, 12, 4, 30, 4, 6, 2, 28, 2, 6, 6, 127, 4, 12, 2, 14, 4, 12, 2, 45, 2, 6, 6, 14, 4, 12, 2, 62

Offset: 1

Author

Ralf Stephan, Nov 27 2010

Keywords

• nonn
• mult
• easy

Comments

The Dirichlet g.f. is the Dirichlet g.f. of A006519 multiplied by zeta(s). - R. J. Mathar, Feb 06 2011

Multiplicative because A006519 is. - Andrew Howroyd, Jul 27 2018

Links

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

Crossrefs

Cf. A000005, A006519, A001227.

Programs

Mathematica

a[n_] := Sum[2^IntegerExponent[d, 2], {d, Divisors[n]}];
Array[a, 80] (* Jean-François Alcover, Feb 16 2020, from PARI *)
f[p_, e_] := If[p==2, 2^(e+1)-1, 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, 2^valuation(d, 2)); \\ Michel Marcus, Mar 27 2015

Formula

a(1) = 1, a(2n) = 2a(n) + A001227(n), a(2n+1) = A000005(2n+1).

Dirichlet g.f.: zeta(s)^2*(1-2^(-s))/(1-2^(-s+1)). - Ralf Stephan, Mar 27 2015

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

Sum_{k=1..n} a(k) ~ n*(log(n)^2/(4*log(2)) + (3/4 - 1/(2*log(2)) + gamma/log(2))*log(n) - 3/4 + log(2)/24 + 1/(2*log(2)) + (3/2 - 1/log(2))*gamma + gamma^2/(2*log(2)) - sg1/log(2)), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Nov 20 2021

Extensions

Title corrected by R. J. Mathar, Feb 06 2011

Terms a(61) and beyond from Andrew Howroyd, Jul 27 2018