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
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 24 2021
STATUS
approved