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%I #20 Nov 04 2023 02:59:38
%S 1,14,64,160,368,592,1104,1520,2400,3056,4640,5264,7824,8736,11776,
%T 13216,17984,18384,25344,26080,33312,35120,45584,44320,58480,58512,
%U 72000,73200,92624,86848,113520,110144,132640,132416,162816,152112,194544,185616,220416
%N a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^4 - floor((n-1)/k)^4).
%F Sum_{k=1..n} a(k) * floor(n/k) = n^4.
%F Sum_{k=1..n} a(k) = A082540(n).
%F G.f.: Sum_{k >= 1} mu(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^4.
%t a[n_] := Sum[MoebiusMu[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^4), {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 24 2021 *)
%o (PARI) a(n) = sum(k=1, n, moebius(k)*((n\k)^4-((n-1)\k)^4));
%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^4))
%Y Cf. A000583, A082540, A140434, A344596.
%K nonn
%O 1,2
%A _Seiichi Manyama_, May 24 2021