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

1,3

Links

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

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A171092 A141256 A122921 * A108128 A081326 A277879

Adjacent sequences: A174270 A174271 A174272 * A174274 A174275 A174276

Keyword

nonn,mult

Author

Ralf Stephan, Nov 27 2010

Extensions

More terms from Antti Karttunen, Sep 27 2018

Status

approved