A367865 a(n) = Sum_{d|n} d * phi(d) * mu(d)^2.

1, 3, 7, 3, 21, 21, 43, 3, 7, 63, 111, 21, 157, 129, 147, 3, 273, 21, 343, 63, 301, 333, 507, 21, 21, 471, 7, 129, 813, 441, 931, 3, 777, 819, 903, 21, 1333, 1029, 1099, 63, 1641, 903, 1807, 333, 147, 1521, 2163, 21, 43, 63, 1911, 471, 2757, 21, 2331, 129, 2401, 2439

Offset: 1

Author

Wesley Ivan Hurt, Dec 03 2023

Keywords

• nonn
• easy
• mult

Comments

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

Links

• N. J. A. Sloane, Transforms.

Crossrefs

Cf. A000010 (phi), A007947 (rad), A008966 (mu^2), A202535.

Programs

Mathematica

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

PARI

a(n) = sumdiv(n, d, if (issquarefree(d), d*eulerphi(d))); \\ Michel Marcus, Dec 04 2023

Python

from math import prod
from sympy import primefactors
def A367865(n): return prod(p*(p-1)+1 for p in primefactors(n)) # Chai Wah Wu, Dec 05 2023

Formula

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

Sum_{k=1..n} a(k) ~ c * n^3/3, where c = Product_{p prime} (1 - 2/(1+p+p^2)) = 0.51478027457383523467921514707014858470711969900467102074735896602342984... - Vaclav Kotesovec, Dec 05 2023