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

1,2

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)

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

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 *)

Prog

(PARI) a(n) = sumdiv(n, d, 2^valuation(d, 2)); \\ Michel Marcus, Mar 27 2015

Crossrefs

Cf. A000005, A006519, A001227.

Sequence in context: A193574 A209639 A366161 * A361145 A175920 A200593

Adjacent sequences: A174235 A174236 A174237 * A174239 A174240 A174241

Keyword

nonn,mult,easy

Author

Ralf Stephan, Nov 27 2010

Extensions

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

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

Status

approved