A357938 Inverse Moebius transform of n * 2^omega(n).

1, 5, 7, 13, 11, 35, 15, 29, 25, 55, 23, 91, 27, 75, 77, 61, 35, 125, 39, 143, 105, 115, 47, 203, 61, 135, 79, 195, 59, 385, 63, 125, 161, 175, 165, 325, 75, 195, 189, 319, 83, 525, 87, 299, 275, 235, 95, 427, 113, 305, 245, 351, 107, 395, 253, 435, 273, 295, 119, 1001

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

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

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

Dirichlet inverse equals Dirichlet convolution of A298473 and A008683.

Sum_{k=1..n} a(k) ~ n^2 * (log(n)/2 + gamma - 1/4 - 3*zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 03 2025

a(n) = Sum_{d|n} A007913(d)*A001615(n/d). - Ridouane Oudra, Jul 03 2025

a(n) = Sum_{d|n} mobius(d)^2*d*sigma(n/d). - Ridouane Oudra, Jul 24 2025

Mathematica

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 *)

Prog

(PARI) a(n) = sumdiv(n, d, d * 2^omega(d)); \\ Michel Marcus, Oct 31 2022
(Python)
from math import prod
from sympy import factorint
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

Crossrefs

Cf. A001221, A008683, A298473, A007913, A001615, A008966, A000203.

Sequence in context: A074755 A354200 A002659 * A164122 A166163 A249801

Adjacent sequences: A357935 A357936 A357937 * A357939 A357940 A357941

Keyword

nonn,easy,mult

Author

Werner Schulte, Oct 24 2022

Status

approved