A338655 a(n) = Sum_{d|n} phi(d) * binomial(d+n/d-1, d).

1, 3, 5, 9, 9, 22, 13, 32, 35, 53, 21, 121, 25, 96, 177, 166, 33, 297, 37, 491, 417, 218, 45, 1002, 549, 297, 705, 1375, 57, 2418, 61, 1640, 1405, 491, 3887, 4659, 73, 606, 2233, 8156, 81, 8989, 85, 6189, 11955, 872, 93, 16550, 10387, 12927, 4757, 11111, 105, 22392, 25757

Offset

1,2

Links

Table of n, a(n) for n=1..55.

Formula

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

If p is prime, a(p) = 2*p - 1.

From Richard L. Ollerton, May 07 2021: (Start)

a(n) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 1,n/gcd(n,k)).

a(n) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 1,gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

Mathematica

a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[# + n/# - 1, #] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)

Prog

(PARI) a(n) = sumdiv(n, d, eulerphi(d)*binomial(d+n/d-1, d));
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-x^k)^(k+1)))

Crossrefs

Cf. A000010, A156834, A343517, A338658.

Sequence in context: A062949 A166651 A081500 * A030365 A316520 A153710

Adjacent sequences: A338652 A338653 A338654 * A338656 A338657 A338658

Keyword

nonn

Author

Seiichi Manyama, Apr 22 2021

Status

approved