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
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
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