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).

1, 7, 23, 64, 130, 287, 468, 864, 1335, 2156, 3013, 4790, 6200, 9072, 11972, 16440, 20365, 28209, 33667, 45014, 54192, 68853, 80752, 104964, 119279, 148778, 172629, 211252, 237364, 295288, 324662, 394368, 442133, 522403, 578385, 696624, 749434, 884443, 975250, 1136476

Offset: 1

Author

Seiichi Manyama, Apr 17 2021

Keywords

• nonn

Links

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

Crossrefs

Column 5 of A343516.

Cf. A000010, A343499.

Programs

Mathematica

a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + 4, 5] &]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)

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))))));

PARI

a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+4, 5));

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^6))

Formula

a(n) = Sum_{d|n} phi(n/d) * binomial(d+4, 5).

G.f.: Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^6.

Sum_{k=1..n} a(k) ~ 21*zeta(5)*n^6 / (16*Pi^6). - Vaclav Kotesovec, May 23 2021