A357938 - OEIS

%I #26 Nov 01 2022 07:14:44

%S 1,5,7,13,11,35,15,29,25,55,23,91,27,75,77,61,35,125,39,143,105,115,

%T 47,203,61,135,79,195,59,385,63,125,161,175,165,325,75,195,189,319,83,

%U 525,87,299,275,235,95,427,113,305,245,351,107,395,253,435,273,295,119,1001

%N Inverse Moebius transform of n * 2^omega(n).

%F Multiplicative with a(p^e) = 2 * (p^(e+1)-1) / (p-1) - 1 for prime p and e >= 0.

%F Dirichlet g.f.: (zeta(s-1))^2 * zeta(s) / zeta(2*s-2).

%F Dirichlet inverse equals Dirichlet convolution of A298473 and A008683.

%t f[p_, e_] := 2*(p^(e + 1) - 1)/(p - 1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* _Amiram Eldar_, Oct 24 2022 *)

%o (PARI) a(n) = sumdiv(n, d, d * 2^omega(d)); \\ _Michel Marcus_, Oct 31 2022

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A357938(n): return prod(((p**(e+1)-1)//(p-1)<<1)-1 for p, e in factorint(n).items()) # _Chai Wah Wu_, Oct 31 2022

%Y Cf. A001221, A008683, A298473.

%K nonn,easy,mult

%O 1,2

%A _Werner Schulte_, Oct 24 2022