A174238 - OEIS

%I #36 Nov 20 2021 06:23:36

%S 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,

%T 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,

%U 2,6,6,127,4,12,2,14,4,12,2,45,2,6,6,14,4,12,2,62

%N Inverse Moebius transform of even part of n (A006519).

%C The Dirichlet g.f. is the Dirichlet g.f. of A006519 multiplied by zeta(s). - _R. J. Mathar_, Feb 06 2011

%C Multiplicative because A006519 is. - _Andrew Howroyd_, Jul 27 2018

%H Amiram Eldar, <a href="/A174238/b174238.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Andrew Howroyd)

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

%F Dirichlet g.f.:  zeta(s)^2*(1-2^(-s))/(1-2^(-s+1)). - _Ralf Stephan_, Mar 27 2015

%F Multiplicative with a(2^e) = 2^(e+1)-1, and a(p^e) = e+1 for p > 2. - _Amiram Eldar_, Sep 30 2020

%F 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

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

%t Array[a, 80] (* _Jean-François Alcover_, Feb 16 2020, from PARI *)

%t 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 *)

%o (PARI) a(n) = sumdiv(n, d, 2^valuation(d, 2)); \\ _Michel Marcus_, Mar 27 2015

%Y Cf. A000005, A006519, A001227.

%K nonn,mult,easy

%O 1,2

%A _Ralf Stephan_, Nov 27 2010

%E Title corrected by _R. J. Mathar_, Feb 06 2011

%E Terms a(61) and beyond from _Andrew Howroyd_, Jul 27 2018