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

1,2

Links

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

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)

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 *)

Prog

(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

Crossrefs

Cf. A000005, A002144, A002145, A002654.

Sequence in context: A355255 A336692 A332434 * A395179 A324338 A047679

Adjacent sequences: A262206 A262207 A262208 * A262210 A262211 A262212

Keyword

nonn,mult,easy

Author

R. J. Mathar, Sep 15 2015

Status

approved