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A343553 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n = n} gcd(x_1, x_2, ... , x_n).

%I A343553 #21 Jun 12 2021 12:50:12
%S A343553 1,3,8,26,74,287,930,3572,12966,49379,184766,710712,2704168,10427822,
%T A343553 40123208,155289768,601080406,2334740919,9075135318,35352194658,
%U A343553 137846990678,538302226835,2104098963742,8233721100024,32247603765020,126412458921072,495918569262798
%N A343553 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n = n} gcd(x_1, x_2, ... , x_n).
%H A343553 Seiichi Manyama, <a href="/A343553/b343553.txt">Table of n, a(n) for n = 1..1000</a>
%F A343553 a(n) = A343516(n,n-1).
%F A343553 a(n) = Sum_{d|n} phi(n/d) * binomial(d+n-2, n-1).
%F A343553 a(n) = [x^n] Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^n.
%F A343553 a(n) ~ 2^(2*n - 2) / sqrt(Pi*n). - _Vaclav Kotesovec_, May 23 2021
%e A343553 a(3) = gcd(1,1,3) + gcd(1,2,3) + gcd(1,3,3) + gcd(2,2,3) + gcd(2,3,3) + gcd(3,3,3) = 1 + 1 + 1 + 1 + 1 + 3 = 8.
%t A343553 a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + n - 2, n-1] &]; Array[a, 30] (* _Amiram Eldar_, Apr 25 2021 *)
%o A343553 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+n-2, n-1));
%Y A343553 Cf. A000010, A332470, A332508, A343516, A343517, A343547.
%K A343553 nonn
%O A343553 1,2
%A A343553 _Seiichi Manyama_, Apr 19 2021