A367866 - OEIS

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

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OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..54.

N. J. A. Sloane, Transforms.

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

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.

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