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%I A344599 #13 May 24 2021 11:54:02
%S A344599 1,8,21,46,65,126,133,224,261,364,341,618,481,722,837,1000,833,1404,
%T A344599 1045,1718,1641,1798,1541,2760,2065,2516,2673,3346,2465,4410,2821,
%U A344599 4256,4041,4312,4469,6462,4033,5390,5637,7504,4961,8532,5461,8186,8613,7906,6533,11736,7861,10640,9621
%N A344599 a(n) = Sum_{k=1..n} phi(k) * (floor(n/k)^3 - floor((n-1)/k)^3).
%F A344599 Sum_{k=1..n} a(k) = A344522(n).
%F A344599 G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3.
%t A344599 a[n_] := Sum[EulerPhi[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^3), {k, 1, n}]; Array[a, 40] (* _Amiram Eldar_, May 24 2021 *)
%o A344599 (PARI) a(n) = sum(k=1, n, eulerphi(k)*((n\k)^3-((n-1)\k)^3));
%o A344599 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3))
%Y A344599 Cf. A344522, A344598, A344600.
%K A344599 nonn
%O A344599 1,2
%A A344599 _Seiichi Manyama_, May 24 2021

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