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A343519 a(n) = Sum_{1 <= x_1 <= x_2 <= x_3 <= x_4 <= x_5 <= n} gcd(x_1, x_2, x_3 , x_4, x_5, n).

%I A343519 #17 May 23 2021 08:43:57
%S A343519 1,7,23,64,130,287,468,864,1335,2156,3013,4790,6200,9072,11972,16440,
%T A343519 20365,28209,33667,45014,54192,68853,80752,104964,119279,148778,
%U A343519 172629,211252,237364,295288,324662,394368,442133,522403,578385,696624,749434,884443,975250,1136476
%N A343519 a(n) = Sum_{1 <= x_1 <= x_2 <= x_3 <= x_4 <= x_5 <= n} gcd(x_1, x_2, x_3 , x_4, x_5, n).
%H A343519 Vaclav Kotesovec, <a href="/A343519/b343519.txt">Table of n, a(n) for n = 1..10000</a>
%F A343519 a(n) = Sum_{d|n} phi(n/d) * binomial(d+4, 5).
%F A343519 G.f.: Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^6.
%F A343519 Sum_{k=1..n} a(k) ~ 21*zeta(5)*n^6 / (16*Pi^6). - _Vaclav Kotesovec_, May 23 2021
%t A343519 a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + 4, 5] &]; Array[a, 50] (* _Amiram Eldar_, Apr 18 2021 *)
%o A343519 (PARI) a(n) = sum(a=1, n, sum(b=1, a, sum(c=1, b, sum(d=1, c, sum(e=1, d, gcd(gcd(gcd(gcd(gcd(n, a), b), c), d), e))))));
%o A343519 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+4, 5));
%o A343519 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^6))
%Y A343519 Column 5 of A343516.
%Y A343519 Cf. A000010, A343499.
%K A343519 nonn
%O A343519 1,2
%A A343519 _Seiichi Manyama_, Apr 17 2021