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A345230 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n <= n} gcd(x_1, x_2, ..., x_n). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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)));

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

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Last modified September 28 10:48 EDT 2021. Contains 347714 sequences. (Running on oeis4.)