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A383783 a(n) = Sum_{k=1..2^n} mu(k) * (floor(2^n/k)^4 - floor((2^n-1)/k)^4).

%I A383783 #19 May 10 2025 11:58:28
%S A383783 1,14,160,1520,13216,110144,899200,7266560,58425856,468583424,
%T A383783 3753379840,30045900800,240442679296,1923843375104,15391954862080,
%U A383783 123140470538240,985143091265536,7881222038749184,63050085546065920,504401921315962880,4035220318323736576
%N A383783 a(n) = Sum_{k=1..2^n} mu(k) * (floor(2^n/k)^4 - floor((2^n-1)/k)^4).
%H A383783 Chai Wah Wu, <a href="/A383783/b383783.txt">Table of n, a(n) for n = 0..48</a>
%F A383783 a(n) = A344597(2^n) = A082540(2^n) - A082540(2^n-1).
%t A383783 a[n_]:=Sum[MoebiusMu[k]*(Floor[2^n/k]^4-Floor[(2^n-1)/k]^4),{k,2^n}]; Array[a,21,0] (* _James C. McMahon_, May 10 2025 *)
%o A383783 (Python)
%o A383783 from functools import lru_cache
%o A383783 @lru_cache(maxsize=None)
%o A383783 def A082540(n):
%o A383783     if n == 0:
%o A383783         return 0
%o A383783     c, j = 1, 2
%o A383783     k1 = n//j
%o A383783     while k1 > 1:
%o A383783         j2 = n//k1 + 1
%o A383783         c += (j2-j)*A082540(k1)
%o A383783         j, k1 = j2, n//j2
%o A383783     return n*(n**3-1)-c+j
%o A383783 def A383783(n): return A082540(m:=1<<n)-A082540(m-1)
%o A383783 (PARI) a(n) = sum(k=1, 2^n, moebius(k) * ((2^n\k)^4 - ((2^n-1)\k)^4)); \\ _Michel Marcus_, May 10 2025
%Y A383783 Cf. A082540, A344597.
%K A383783 nonn
%O A383783 0,2
%A A383783 _Chai Wah Wu_, May 09 2025