A343553 - OEIS

%I #21 Jun 12 2021 12:50:12

%S 1,3,8,26,74,287,930,3572,12966,49379,184766,710712,2704168,10427822,

%T 40123208,155289768,601080406,2334740919,9075135318,35352194658,

%U 137846990678,538302226835,2104098963742,8233721100024,32247603765020,126412458921072,495918569262798

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

%H Seiichi Manyama, <a href="/A343553/b343553.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A343516(n,n-1).

%F a(n) = Sum_{d|n} phi(n/d) * binomial(d+n-2, n-1).

%F a(n) = [x^n] Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^n.

%F a(n) ~ 2^(2*n - 2) / sqrt(Pi*n). - _Vaclav Kotesovec_, May 23 2021

%e 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 a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + n - 2, n-1] &]; Array[a, 30] (* _Amiram Eldar_, Apr 25 2021 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+n-2, n-1));

%Y Cf. A000010, A332470, A332508, A343516, A343517, A343547.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Apr 19 2021