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A365932 a(n) = the number of cubes (of integers > 0) that have bit length n.

%I A365932 #36 Nov 01 2023 10:02:00
%S A365932 1,0,0,1,1,0,2,1,1,3,2,3,5,5,6,9,10,13,17,21,26,34,42,52,67,84,105,
%T A365932 134,167,211,267,335,422,533,670,845,1065,1341,1690,2130,2682,3380,
%U A365932 4259,5365,6760,8518,10730,13520,17035,21461,27040,34069,42923,54080,68137,85847
%N A365932 a(n) = the number of cubes (of integers > 0) that have bit length n.
%C A365932 Number of cubes in the range: 2^(n-1) <= k^3 < 2^n-1.
%C A365932 There is no need to include 2^n-1 because it is a Mersenne number and it cannot be a power anyway.
%F A365932 a(n) = floor((2^n-1)^(1/3)) - floor((2^(n-1)-1)^(1/3)) for n > 0.
%F A365932 Limit_{n->oo} a(n)/a(n-1) = 2^(1/3) = A002580.
%e A365932 For n = 13; a(n) = 5; following 5 cubes have a bit length of 13: 16^3, 17^3, 18^3, 19^3 and 20^3.
%t A365932 a[n_] := Floor[Surd[2^n-1, 3]] - Floor[Surd[2^(n-1)-1, 3]]; Array[a, 56] (* _Amiram Eldar_, Oct 30 2023 *)
%o A365932 (Python)
%o A365932 from sympy import integer_nthroot
%o A365932 def A365932(n):
%o A365932     return integer_nthroot(2**n-1, 3)[0] - integer_nthroot(2**(n-1)-1, 3)[0]
%o A365932 print([A365932(n) for n in range(1,57)])
%Y A365932 Cf. A004632.
%Y A365932 Cf. A017981 (partial sums).
%Y A365932 Cf. A002580, A126726, A365930.
%K A365932 nonn,easy,base
%O A365932 1,7
%A A365932 _Karl-Heinz Hofmann_, Oct 05 2023