A262209 Inverse Möbius transform of A002654.

1, 2, 1, 3, 3, 2, 1, 4, 2, 6, 1, 3, 3, 2, 3, 5, 3, 4, 1, 9, 1, 2, 1, 4, 6, 6, 2, 3, 3, 6, 1, 6, 1, 6, 3, 6, 3, 2, 3, 12, 3, 2, 1, 3, 6, 2, 1, 5, 2, 12, 3, 9, 3, 4, 3, 4, 1, 6, 1, 9, 3, 2, 2, 7, 9, 2, 1, 9, 1, 6, 1, 8, 3, 6, 6, 3, 1, 6, 1, 15, 3, 6, 1, 3, 9

Offset: 1

Author

R. J. Mathar, Sep 15 2015

Keywords

• nonn
• mult
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A002144, A002145, A002654.

Programs

Mathematica

f[2, e_] := e + 1; f[p_, e_] := If[Mod[p, 4] == 1, (e + 1)*(e + 2)/2, Floor[e/2] + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 01 2025 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, f[i, 2] + 1, if(f[i, 1] % 4 == 1, (f[i, 2]+1)*(f[i, 2]+2)/2, f[i, 2]\2 + 1)));} \\ Amiram Eldar, Feb 01 2025

Formula

G.f.: Sum_{k>=1} tau(k)*x^k/(1 + x^(2*k)), where tau = A000005. - Ilya Gutkovskiy, Sep 14 2019

From Amiram Eldar, Feb 01 2025: (Start)

a(n) = Sum_{d|n} A002654(d).

Multiplicative with a(p^e) = e+1 if p = 2, a(p^e) = floor(e/2) + 1 if p == 3 (mod 4) (A002145), and a(p^e) = (e+1)*(e+2)/2 if p == 1 (mod 4) (A002144). (End)