OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
FORMULA
Inverse Möbius transform of A001227, the number of odd divisors of n. Row sums of triangle A133699.
Dirichlet g.f. (zeta(s))^3*(1-1/2^s). - R. J. Mathar, Feb 07 2011
a(n) = Sum_{d|n} A001227(d). - Antti Karttunen, Sep 27 2018
Sum_{k=1..n} a(k) ~ n/4 * (log(n)^2 + (6*g - 2 + 2*log(2))*log(n) + 2 + 6*g^2 - log(2)^2 - 2*log(2) + 6*g*(log(2) - 1) - 6*sg1), where g is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant A082633. - Vaclav Kotesovec, Feb 02 2019
G.f.: Sum_{k>=1} tau(k)*x^k/(1 - x^(2*k)), where tau = A000005. - Ilya Gutkovskiy, Sep 13 2019
Multiplicative with a(2^e) = e+1, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - Amiram Eldar, Oct 28 2023
EXAMPLE
MATHEMATICA
f[p_, e_] := (e+1)*(e+2)/2; f[2, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 28 2023 *)
PROG
(PARI) A133700(n) = sumdiv(n, d, numdiv(d>>valuation(d, 2))); \\ Antti Karttunen, Sep 27 2018
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Gary W. Adamson, Sep 21 2007
EXTENSIONS
More terms from R. J. Mathar, Jan 19 2009
Second, equivalent formula added to the definition by Antti Karttunen, Sep 27 2018
STATUS
approved