A344596 - OEIS

%I #34 Nov 04 2023 03:00:03

%S 1,6,18,30,60,66,126,132,198,204,330,276,468,414,552,552,816,630,1026,

%T 840,1116,1050,1518,1128,1740,1476,1890,1692,2436,1704,2790,2256,2820,

%U 2544,3384,2556,3996,3186,3960,3408,4920,3420,5418,4260,5112,4686,6486,4560,6930,5340,6816

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

%H Seiichi Manyama, <a href="/A344596/b344596.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

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

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

%o (PARI) a(n) = sum(k=1, n, moebius(k)*((n\k)^3-((n-1)\k)^3));

%o (PARI) a(n) = if(n<2, n, 3*sumdiv(n, d, moebius(n/d)*(d-1)*d));

%o (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))

%o (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))

%Y Essentially 6*A102309 and 6*A326419.

%Y Cf. A000578, A071778, A140434, A344597.

%K nonn

%O 1,2

%A _Seiichi Manyama_, May 24 2021