

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
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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*(12^(s))/(12^(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] (* JeanFrançois 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



