OFFSET
0,3
COMMENTS
In general, for m >= 1, Sum_{k=1..n} psi_m(k) = Sum_{k=1..n} mu(k)^2 * (Bernoulli(m+1, 1+floor(n/k)) - Bernoulli(m+1, 1)) / (m+1), where mu(k) is the Moebius function and Bernoulli(n,x) are the Bernoulli polynomials.
In general, for m >= 1, Sum_{k=1..n} psi_m(k) ~ n^(m+1) * zeta(m+1) / ((m+1) * zeta(2*(m+1))).
LINKS
FORMULA
a(n) = Sum_{k=1..n} A065958(k).
a(n) ~ n^3 * zeta(3) / (3*zeta(6)).
a(n) = Sum_{k=1..n} mu(k)^2 * Bernoulli(3, 1+floor(n/k)) / 3.
MATHEMATICA
a[n_] := Sum[MoebiusMu[k]^2 * BernoulliB[3, 1 + Floor[n/k]], {k, 1, n}]/3; Array[a, 50, 0] (* Amiram Eldar, Nov 23 2018 *)
PROG
(PARI) a(n) = sum(k=1, n, moebius(k)^2 * ((n\k)^3/3 + (n\k)^2/2 + (n\k)/6));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Daniel Suteu, Nov 22 2018
STATUS
approved