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

1,2

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).

Links

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

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

From Aloe Poliszuk, Nov 12 2025: (Start)

a(n) = Sum_{d|n} (n/d)^3 * A023900(d).

a(n) = Sum_{d|n} d * A007434(d). (End)

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 *)

Prog

(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
(PARI) a(n) = sumdiv(n, d, d*moebius(d)*sigma(n/d, 3)); \\ Aloe Poliszuk, Nov 12 2025

Crossrefs

Column k=2 of A372968.

Cf. A000010, A000056, A000578, A001158, A007434, A023900, A055615, A057660.

Cf. A372962, A372964.

Sequence in context: A155313 A213390 A071778 * A155250 A155260 A155244

Adjacent sequences: A350153 A350154 A350155 * A350157 A350158 A350159

Keyword

nonn,easy,mult

Author

Werner Schulte, Jan 19 2022

Status

approved