A262209 - OEIS

%I #20 Feb 01 2025 08:50:44

%S 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,

%T 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,

%U 9,1,6,1,8,3,6,6,3,1,6,1,15,3,6,1,3,9

%N Inverse Möbius transform of A002654.

%H Amiram Eldar, <a href="/A262209/b262209.txt">Table of n, a(n) for n = 1..10000</a>

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

%F From _Amiram Eldar_, Feb 01 2025: (Start)

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

%F 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)

%t 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 *)

%o (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

%Y Cf. A000005, A002144, A002145, A002654.

%K nonn,mult,easy

%O 1,2

%A _R. J. Mathar_, Sep 15 2015