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A343518 a(n) = Sum_{1 <= x_1 <= x_2 <= x_3 <= x_4 <= n} gcd(x_1, x_2, x_3 , x_4, n).

%I A343518 #19 May 23 2021 08:43:24
%S A343518 1,6,17,42,74,153,216,379,531,809,1011,1605,1832,2626,3268,4304,4861,
%T A343518 6798,7333,9878,11148,13711,14972,19985,20775,25643,28503,34517,35988,
%U A343518 46162,46406,57092,61077,70986,75099,92520,91426,108693,115774,135491,135791,165719,163227,193437
%N A343518 a(n) = Sum_{1 <= x_1 <= x_2 <= x_3 <= x_4 <= n} gcd(x_1, x_2, x_3 , x_4, n).
%H A343518 Vaclav Kotesovec, <a href="/A343518/b343518.txt">Table of n, a(n) for n = 1..10000</a>
%F A343518 a(n) = Sum_{d|n} phi(n/d) * binomial(d+3, 4).
%F A343518 G.f.: Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^5.
%F A343518 Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / (10800*zeta(5)). - _Vaclav Kotesovec_, May 23 2021
%t A343518 a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + 3, 4] &]; Array[a, 50] (* _Amiram Eldar_, Apr 18 2021 *)
%o A343518 (PARI) a(n) = sum(a=1, n, sum(b=1, a, sum(c=1, b, sum(d=1, c, gcd(gcd(gcd(gcd(n, a), b), c), d)))));
%o A343518 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+3, 4));
%o A343518 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^5))
%Y A343518 Column 4 of A343516.
%Y A343518 Cf. A000010, A343498.
%K A343518 nonn
%O A343518 1,2
%A A343518 _Seiichi Manyama_, Apr 17 2021