A345230 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n <= n} gcd(x_1, x_2, ..., x_n).

0, 1, 4, 13, 44, 140, 512, 1782, 6652, 24682, 93599, 354341, 1359470, 5210328, 20098886, 77621774, 300797854, 1167164438, 4539201401, 17674941735, 68933414989, 269143872226, 1052114789548, 4116808923486, 16124224585644, 63205911146740, 247961982954952

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

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

a(n) ~ 2^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 11 2021

a(n) = Sum_{k=1..n} phi(k) * binomial(floor(n/k)+n-1,n). - Seiichi Manyama, Sep 13 2024

Maple

a:= n-> coeff(series((1/(1-x))* add(numtheory[phi](k)
         *x^k/(1-x^k)^n, k=1..n), x, n+1), x, n):
seq(a(n), n=0..26);  # Alois P. Heinz, Jun 11 2021

Mathematica

a[n_] := Sum[DivisorSum[k, EulerPhi[k/#] * Binomial[n + # - 2, n - 1] &], {k, 1, n}]; Array[a, 30, 0] (* Amiram Eldar, Jun 11 2021 *)

Prog

(PARI) a(n) = sum(k=1, n, sumdiv(k, d, eulerphi(k/d)*binomial(d+n-2, n-1)));
(PARI) a(n) = sum(k=1, n, eulerphi(k)*binomial(n\k+n-1, n)); \\ Seiichi Manyama, Sep 13 2024

Crossrefs

Main diagonal of A345229.

Cf. A343517, A343553, A344525.

Sequence in context: A286175 A252831 A219708 * A117882 A257674 A027123

Adjacent sequences: A345227 A345228 A345229 * A345231 A345232 A345233

Keyword

nonn

Author

Seiichi Manyama, Jun 11 2021

Status

approved