A367866 a(n) = Sum_{d|n} d * sigma(d) * mu(d)^2.

1, 7, 13, 7, 31, 91, 57, 7, 13, 217, 133, 91, 183, 399, 403, 7, 307, 91, 381, 217, 741, 931, 553, 91, 31, 1281, 13, 399, 871, 2821, 993, 7, 1729, 2149, 1767, 91, 1407, 2667, 2379, 217, 1723, 5187, 1893, 931, 403, 3871, 2257, 91, 57, 217, 3991, 1281, 2863, 91

Offset

1,2

Comments

Inverse Möbius transform of n * sigma(n) * mu(n)^2.

Links

Aloe Poliszuk, Table of n, a(n) for n = 1..10000

N. J. A. Sloane, Transforms.

Wikipedia, Littlewood polynomial.

Formula

Multiplicative with a(p^e) = p^2 + p + 1. - Amiram Eldar, Dec 04 2023

Sum_{k=1..n} a(k) ~ n^3/3. - Vaclav Kotesovec, Dec 05 2023

From Aloe Poliszuk, Oct 15 2025: (Start)

a(n) = sigma(rad(n)^2), where sigma = A000203 and rad = A007947.

a(n) = Sum_{d|n} d*mu(d)^2 * Sum_{k|d} k*mu(k)^2. (End)

Mathematica

Table[Sum[d*DivisorSigma[1, d]*MoebiusMu[d]^2, {d, Divisors[n]}], {n, 100}]

Prog

(PARI) a(n) = sumdiv(n, d, if (issquarefree(d), d*sigma(d))); \\ Michel Marcus, Dec 04 2023
(Python)
from math import prod
from sympy import primefactors
def A367866(n): return prod(p*(p+1)+1 for p in primefactors(n)) # Chai Wah Wu, Dec 05 2023

Crossrefs

Cf. A000203 (sigma), A008966 (mu^2), A343442, A007947 (rad), A048250, A202535.

Related arithmetic functions: A367866, A389775, A389977, A389978.

Sequence in context: A046163 A130770 A158622 * A369717 A365346 A215990

Adjacent sequences: A367863 A367864 A367865 * A367867 A367868 A367869

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, Dec 03 2023

Status

approved