A343521 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).

1, 9, 38, 130, 334, 846, 1722, 3572, 6513, 11806, 19458, 32948, 50400, 79290, 117092, 174256, 245173, 354249, 480718, 670420, 891690, 1203578, 1560802, 2076496, 2630915, 3416352, 4285152, 5461348, 6724548, 8490884, 10295502, 12798224, 15420213, 18888861

Offset

1,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Formula

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

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

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

Mathematica

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

Prog

(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+6, 7));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^8))

Crossrefs

Column 7 of A343516.

Cf. A000010, A343509.

Sequence in context: A071229 A071238 A213583 * A050854 A053181 A348545

Adjacent sequences: A343518 A343519 A343520 * A343522 A343523 A343524

Keyword

nonn

Author

Seiichi Manyama, Apr 17 2021

Status

approved