A344596 a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^3 - floor((n-1)/k)^3).

1, 6, 18, 30, 60, 66, 126, 132, 198, 204, 330, 276, 468, 414, 552, 552, 816, 630, 1026, 840, 1116, 1050, 1518, 1128, 1740, 1476, 1890, 1692, 2436, 1704, 2790, 2256, 2820, 2544, 3384, 2556, 3996, 3186, 3960, 3408, 4920, 3420, 5418, 4260, 5112, 4686, 6486, 4560, 6930, 5340, 6816

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

Sum_{k=1..n} a(k) * floor(n/k) = n^3.

Sum_{k=1..n} a(k) = A071778(n).

a(n) = 3 * Sum_{d|n} mu(n/d) * (d-1) * d for n > 1.

G.f.: Sum_{k >= 1} mu(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3.

G.f.: x + 6 * Sum_{k>=1} mu(k) * x^(2*k)/(1 - x^k)^3.

Mathematica

a[n_] := Sum[MoebiusMu[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^3), {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 24 2021 *)

Prog

(PARI) a(n) = sum(k=1, n, moebius(k)*((n\k)^3-((n-1)\k)^3));
(PARI) a(n) = if(n<2, n, 3*sumdiv(n, d, moebius(n/d)*(d-1)*d));
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3))
(PARI) my(N=66, x='x+O('x^N)); Vec(x+6*sum(k=1, N, moebius(k)*x^(2*k)/(1-x^k)^3))

Crossrefs

Essentially 6*A102309 and 6*A326419.

Cf. A000578, A071778, A140434, A344597.

Sequence in context: A335908 A096286 A256256 * A280802 A124353 A232336

Adjacent sequences: A344593 A344594 A344595 * A344597 A344598 A344599

Keyword

nonn

Author

Seiichi Manyama, May 24 2021

Status

approved