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A344597 a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^4 - floor((n-1)/k)^4).

%I A344597 #24 May 10 2025 02:39:43
%S A344597 1,14,64,160,368,592,1104,1520,2400,3056,4640,5264,7824,8736,11776,
%T A344597 13216,17984,18384,25344,26080,33312,35120,45584,44320,58480,58512,
%U A344597 72000,73200,92624,86848,113520,110144,132640,132416,162816,152112,194544,185616,220416
%N A344597 a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^4 - floor((n-1)/k)^4).
%F A344597 Sum_{k=1..n} a(k) * floor(n/k) = n^4.
%F A344597 Sum_{k=1..n} a(k) = A082540(n).
%F A344597 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 A344597 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 A344597 (PARI) a(n) = sum(k=1, n, moebius(k)*((n\k)^4-((n-1)\k)^4));
%o A344597 (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))
%o A344597 (Python)
%o A344597 from functools import lru_cache
%o A344597 @lru_cache(maxsize=None)
%o A344597 def A082540(n):
%o A344597     if n == 0:
%o A344597         return 0
%o A344597     c, j = 1, 2
%o A344597     k1 = n//j
%o A344597     while k1 > 1:
%o A344597         j2 = n//k1 + 1
%o A344597         c += (j2-j)*A082540(k1)
%o A344597         j, k1 = j2, n//j2
%o A344597     return n*(n**3-1)-c+j
%o A344597 def A344597(n): return A082540(n)-A082540(n-1) # _Chai Wah Wu_, May 09 2025
%Y A344597 Cf. A000583, A082540, A140434, A344596.
%K A344597 nonn
%O A344597 1,2
%A A344597 _Seiichi Manyama_, May 24 2021