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A174238 Inverse Moebius transform of even part of n (A006519). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A Dirichlet g.f. would be greatly appreciated.

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

MATHEMATICA

a[n_] := Sum[2^IntegerExponent[d, 2], {d, Divisors[n]}];

Array[a, 80] (* Jean-Fran├žois 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: A302714 A193574 A209639 * A175920 A200593 A249112

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

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Last modified August 2 02:53 EDT 2021. Contains 346409 sequences. (Running on oeis4.)