A343553 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n = n} gcd(x_1, x_2, ... , x_n).

1, 3, 8, 26, 74, 287, 930, 3572, 12966, 49379, 184766, 710712, 2704168, 10427822, 40123208, 155289768, 601080406, 2334740919, 9075135318, 35352194658, 137846990678, 538302226835, 2104098963742, 8233721100024, 32247603765020, 126412458921072, 495918569262798

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Formula

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

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

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

a(n) ~ 2^(2*n - 2) / sqrt(Pi*n). - Vaclav Kotesovec, May 23 2021

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A206141 A281972 A355869 * A317852 A306485 A148801

Adjacent sequences: A343550 A343551 A343552 * A343554 A343555 A343556

Keyword

nonn

Author

Seiichi Manyama, Apr 19 2021

Status

approved