A350156 Inverse Moebius transform of A000056.

1, 7, 25, 55, 121, 175, 337, 439, 673, 847, 1321, 1375, 2185, 2359, 3025, 3511, 4897, 4711, 6841, 6655, 8425, 9247, 12145, 10975, 15121, 15295, 18169, 18535, 24361, 21175, 29761, 28087, 33025, 34279, 40777, 37015, 50617, 47887, 54625, 53119, 68881, 58975, 79465, 72655, 81433, 85015

Offset: 1

Author

Werner Schulte, Jan 19 2022

Keywords

• nonn
• easy
• mult

Comments

Let f be an arbitrary arithmetic function. Define the sequence a(f; n) by a(f; n) = Sum_{i=1..n, k=1..n} f(n / gcd(gcd(i,k),n)) for n > 0. Then a(f; n) equals inverse Moebius transform of f(n) * A007434(n) for n > 0; if f is multiplicative then a(f; n) is multiplicative; this sequence uses f(n) = n (see formula section).

Crossrefs

Column k=2 of A372968.

Cf. A000010, A000056, A000578, A001158, A007434, A055615, A057660.

Cf. A372962, A372964.

Programs

Mathematica

f[p_, e_] := p^(3*e) - (p - 1)*(p^(3*e) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Jan 19 2022 *)

Python

from math import prod
from sympy import factorint
def A350156(n): return prod((q:=p**(3*e))-(p-1)*(q-1)//(p**3-1) for p,e in factorint(n).items()) # Chai Wah Wu, Mar 04 2025

Formula

Multiplicative with a(p^e) = p^(3*e) - (p-1) * (p^(3*e) - 1) / (p^3 - 1) for prime p and e >= 0.

Dirichlet g.f.: Sum_{n>0} a(n) / n^s = zeta(s-3) * zeta(s) / zeta(s-1).

a(n) = Sum_{i=1..n, k=1..n} n / gcd(gcd(i,k),n) for n > 0.

Dirichlet convolution with A000010 equals A000578.

Dirichlet convolution of A001158 and A055615.

Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/(360*zeta(3)) = 0.225098... . - Amiram Eldar, Oct 16 2022

a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_2(d^2)/sigma(d^2). - Seiichi Manyama, May 24 2024

a(n) = Sum_{1 <= x_1, x_2 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, n) )^2. - Seiichi Manyama, May 25 2024