A343518 a(n) = Sum_{1 <= x_1 <= x_2 <= x_3 <= x_4 <= n} gcd(x_1, x_2, x_3 , x_4, n).

1, 6, 17, 42, 74, 153, 216, 379, 531, 809, 1011, 1605, 1832, 2626, 3268, 4304, 4861, 6798, 7333, 9878, 11148, 13711, 14972, 19985, 20775, 25643, 28503, 34517, 35988, 46162, 46406, 57092, 61077, 70986, 75099, 92520, 91426, 108693, 115774, 135491, 135791, 165719, 163227, 193437

Offset

1,2

Links

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

Formula

a(n) = Sum_{d|n} phi(n/d) * binomial(d+3, 4).

G.f.: Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^5.

Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / (10800*zeta(5)). - Vaclav Kotesovec, May 23 2021

Mathematica

a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[# + 3, 4] &]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)

Prog

(PARI) a(n) = sum(a=1, n, sum(b=1, a, sum(c=1, b, sum(d=1, c, gcd(gcd(gcd(gcd(n, a), b), c), d)))));
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+3, 4));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^5))

Crossrefs

Column 4 of A343516.

Cf. A000010, A343498.

Sequence in context: A013319 A047861 A370589 * A365409 A171507 A099858

Adjacent sequences: A343515 A343516 A343517 * A343519 A343520 A343521

Keyword

nonn

Author

Seiichi Manyama, Apr 17 2021

Status

approved