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 A344992 #31 Sep 13 2024 11:59:37
%S A344992 1,6,18,44,83,159,249,401,592,867,1163,1655,2122,2796,3594,4594,5579,
%T A344992 7046,8394,10328,12339,14699,17021,20441,23526,27317,31379,36323,
%U A344992 40846,47300,52786,59954,67191,75380,83720,94662,103837,115137,126851,141059,153440
%N A344992 a(n) = Sum_{1 <= i <= j <= k <= m <= n} gcd(i,j,k,m).
%C A344992 In general, if g.f.: 1/(1-x) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^k, where k > 2 and phi is the Euler totient function (A000010), then a(n) ~ zeta(k-1) * n^k / (k! * zeta(k)).
%H A344992 Vaclav Kotesovec, <a href="/A344992/b344992.txt">Table of n, a(n) for n = 1..10000</a>
%F A344992 G.f.: 1/(1-x) * Sum_{k>=1} phi(k) * x^k/(1 - x^k)^4, where phi is the Euler totient function (A000010).
%F A344992 a(n) = Sum_{k=1..n} Sum_{d|k} phi(k/d) * d*(d+1)*(d+2)/6.
%F A344992 a(n) ~ 15 * zeta(3) * n^4 / (4*Pi^4).
%F A344992 a(n) = Sum_{k=1..n} phi(k) * binomial(floor(n/k)+3,4). - _Seiichi Manyama_, Sep 13 2024
%t A344992 Table[Sum[Sum[Sum[Sum[GCD[i, j, k, m], {i, 1, j}], {j, 1, k}], {k, 1, m}], {m, 1, n}], {n, 1, 100}]
%t A344992 nmax = 100; Rest[CoefficientList[Series[1/(1-x) * Sum[EulerPhi[k]*x^k/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
%t A344992 Accumulate[Table[Sum[EulerPhi[n/d] * d*(d+1)*(d+2)/6, {d, Divisors[n]}], {n, 1, 100}]] (* faster *)
%o A344992 (PARI) a(n) = sum(i=1, n, sum(j=i, n, sum(k=j, n, sum(m=k, n, gcd([i, j, k, m]))))); \\ _Michel Marcus_, Jun 06 2021
%Y A344992 Column k=4 of A345229.
%Y A344992 Partial sums of A309323.
%Y A344992 Cf. A000010, A059358, A272718, A309322, A344521.
%K A344992 nonn
%O A344992 1,2
%A A344992 _Vaclav Kotesovec_, Jun 05 2021