A343521 - OEIS

%I #20 May 23 2021 08:30:46

%S 1,9,38,130,334,846,1722,3572,6513,11806,19458,32948,50400,79290,

%T 117092,174256,245173,354249,480718,670420,891690,1203578,1560802,

%U 2076496,2630915,3416352,4285152,5461348,6724548,8490884,10295502,12798224,15420213,18888861

%N a(n) = Sum_{1 <= x_1 <= x_2 <= x_3 <= x_4 <= x_5 <= x_6 <= x_7 <= n} gcd(x_1, x_2, x_3 , x_4, x_5, x_6, x_7, n).

%C In general, if m > 1 and a(n) = Sum_{d|n} phi(n/d) * binomial(d + m - 1, m) then Sum_{k=1..n} a(k) ~ zeta(m) * n^(m+1) / ((m+1)! * zeta(m+1)). - _Vaclav Kotesovec_, May 23 2021

%H Vaclav Kotesovec, <a href="/A343521/b343521.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} phi(n/d) * binomial(d+6, 7).

%F G.f.: Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^8.

%F Sum_{k=1..n} a(k) ~ 15*zeta(7)*n^8 / (64*Pi^8). - _Vaclav Kotesovec_, May 23 2021

%t a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + 6, 7] &]; Array[a, 50] (* _Amiram Eldar_, Apr 18 2021 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+6, 7));

%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^8))

%Y Column 7 of A343516.

%Y Cf. A000010, A343509.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Apr 17 2021