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%I #7 Dec 20 2018 23:21:27
%S 0,1,6,16,36,62,112,162,242,332,462,584,784,954,1204,1464,1784,2074,
%T 2524,2886,3406,3906,4516,5046,5846,6496,7346,8156,9156,9998,11298,
%U 12260,13540,14760,16210,17510,19310,20680,22490,24190,26270,27952,30452,32302,34742
%N Partial sums of the Dedekind psi_2(k) function, for 1 <= k <= n.
%C 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.
%C In general, for m >= 1, Sum_{k=1..n} psi_m(k) ~ n^(m+1) * zeta(m+1) / ((m+1) * zeta(2*(m+1))).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dedekind_psi_function">Dedekind psi function</a>
%F a(n) = Sum_{k=1..n} A065958(k).
%F a(n) ~ n^3 * zeta(3) / (3*zeta(6)).
%F a(n) = Sum_{k=1..n} mu(k)^2 * Bernoulli(3, 1+floor(n/k)) / 3.
%t 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 *)
%o (PARI) a(n) = sum(k=1, n, moebius(k)^2 * ((n\k)^3/3 + (n\k)^2/2 + (n\k)/6));
%Y Cf. A001615, A065958, A173290.
%K nonn,easy
%O 0,3
%A _Daniel Suteu_, Nov 22 2018
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