A174273 Inverse Moebius transform of A035263.

1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 4, 3, 2, 3, 2, 4, 4, 2, 2, 4, 3, 2, 4, 4, 2, 4, 2, 3, 4, 2, 4, 6, 2, 2, 4, 4, 2, 4, 2, 4, 6, 2, 2, 6, 3, 3, 4, 4, 2, 4, 4, 4, 4, 2, 2, 8, 2, 2, 6, 4, 4, 4, 2, 4, 4, 4, 2, 6, 2, 2, 6, 4, 4, 4, 2, 6, 5, 2, 2, 8, 4, 2, 4, 4, 2, 6, 4, 4, 4, 2, 4, 6, 2, 3, 6, 6, 2, 4, 2, 4, 8

Offset: 1

Author

Ralf Stephan, Nov 27 2010

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000005, A000035, A000203, A001511, A001620, A035263.

Programs

Mathematica

f[p_, e_] := e + 1 - If[p == 2, Floor[(e + 1)/2], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2022 *)

PARI

A174273(n) = sumdiv(n,d,(valuation(2*d,2)%2)); \\ Antti Karttunen, Sep 27 2018

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,2]+1-if(f[i,1] == 2, floor((f[i,2]+1)/2), 0)); } \\ Amiram Eldar, Nov 13 2022

Formula

a(1) = 1, a(2n) = -a(n) + A000005(2n), a(2n+1) = sigma(2n+1) (A000203).

Dirichlet g.f.: 2^s*(zeta(s))^2/(2^s+1). - R. J. Mathar, Feb 06 2011

From Amiram Eldar, Nov 13 2022: (Start)

Multiplicative with s(2^e) = e+1-floor((e+1)/2) and a(p^e) = e+1 if p>2.

Sum_{k=1..n} a(k) ~ (2/3)*n*log(n) + (2/3)*(2*gamma - 1 + log(2)/2)*n, where gamma is Euler's constant (A001620). (End)

Extensions

More terms from Antti Karttunen, Sep 27 2018