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A343520 a(n) = Sum_{1 <= x_1 <= x_2 <= x_3 <= x_4 <= x_5 <= x_6 <= n} gcd(x_1, x_2, x_3 , x_4, x_5, x_6, n).

%I A343520 #16 May 23 2021 08:44:21
%S A343520 1,8,30,93,214,506,930,1818,3065,5247,8018,13080,18576,28104,39300,
%T A343520 56184,74629,104978,134614,182897,232258,304098,376762,492068,594635,
%U A343520 754941,912384,1137106,1344932,1674374,1947822,2382888,2776997,3337364,3843360,4629687,5245822,6231194
%N A343520 a(n) = Sum_{1 <= x_1 <= x_2 <= x_3 <= x_4 <= x_5 <= x_6 <= n} gcd(x_1, x_2, x_3 , x_4, x_5, x_6, n).
%H A343520 Vaclav Kotesovec, <a href="/A343520/b343520.txt">Table of n, a(n) for n = 1..10000</a>
%F A343520 a(n) = Sum_{d|n} phi(n/d) * binomial(d+5, 6).
%F A343520 G.f.: Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^7.
%F A343520 Sum_{k=1..n} a(k) ~ Pi^6 * n^7 / (4762800*zeta(7)). - _Vaclav Kotesovec_, May 23 2021
%t A343520 a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + 5, 6] &]; Array[a, 50] (* _Amiram Eldar_, Apr 18 2021 *)
%o A343520 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+5, 6));
%o A343520 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^7))
%Y A343520 Column 6 of A343516.
%Y A343520 Cf. A000010, A343508.
%K A343520 nonn
%O A343520 1,2
%A A343520 _Seiichi Manyama_, Apr 17 2021