This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350156 #53 Mar 04 2025 15:21:24
%S A350156 1,7,25,55,121,175,337,439,673,847,1321,1375,2185,2359,3025,3511,4897,
%T A350156 4711,6841,6655,8425,9247,12145,10975,15121,15295,18169,18535,24361,
%U A350156 21175,29761,28087,33025,34279,40777,37015,50617,47887,54625,53119,68881,58975,79465,72655,81433,85015
%N A350156 Inverse Moebius transform of A000056.
%C A350156 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).
%F A350156 Multiplicative with a(p^e) = p^(3*e) - (p-1) * (p^(3*e) - 1) / (p^3 - 1) for prime p and e >= 0.
%F A350156 Dirichlet g.f.: Sum_{n>0} a(n) / n^s = zeta(s-3) * zeta(s) / zeta(s-1).
%F A350156 a(n) = Sum_{i=1..n, k=1..n} n / gcd(gcd(i,k),n) for n > 0.
%F A350156 Dirichlet convolution with A000010 equals A000578.
%F A350156 Dirichlet convolution of A001158 and A055615.
%F A350156 Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/(360*zeta(3)) = 0.225098... . - _Amiram Eldar_, Oct 16 2022
%F A350156 a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_2(d^2)/sigma(d^2). - _Seiichi Manyama_, May 24 2024
%F A350156 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
%t A350156 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 *)
%o A350156 (Python)
%o A350156 from math import prod
%o A350156 from sympy import factorint
%o A350156 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
%Y A350156 Column k=2 of A372968.
%Y A350156 Cf. A000010, A000056, A000578, A001158, A007434, A055615, A057660.
%Y A350156 Cf. A372962, A372964.
%K A350156 nonn,easy,mult
%O A350156 1,2
%A A350156 _Werner Schulte_, Jan 19 2022