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

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, 134, 167, 211, 267, 335, 422, 533, 670, 845, 1065, 1341, 1690, 2130, 2682, 3380, 4259, 5365, 6760, 8518, 10730, 13520, 17035, 21461, 27040, 34069, 42923, 54080, 68137, 85847

Offset

1,7

Comments

Number of cubes in the range: 2^(n-1) <= k^3 < 2^n-1.

There is no need to include 2^n-1 because it is a Mersenne number and it cannot be a power anyway.

Links

Table of n, a(n) for n=1..56.

Formula

a(n) = floor((2^n-1)^(1/3)) - floor((2^(n-1)-1)^(1/3)) for n > 0.

Limit_{n->oo} a(n)/a(n-1) = 2^(1/3) = A002580.

Example

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.

Mathematica

a[n_] := Floor[Surd[2^n-1, 3]] - Floor[Surd[2^(n-1)-1, 3]]; Array[a, 56] (* Amiram Eldar, Oct 30 2023 *)

Prog

(Python)
from sympy import integer_nthroot
def A365932(n):
    return integer_nthroot(2**n-1, 3)[0] - integer_nthroot(2**(n-1)-1, 3)[0]
print([A365932(n) for n in range(1, 57)])

Crossrefs

Cf. A004632.

Cf. A017981 (partial sums).

Cf. A002580, A126726, A365930.

Sequence in context: A291905 A347584 A366398 * A240853 A363094 A319849

Adjacent sequences: A365929 A365930 A365931 * A365933 A365934 A365935

Keyword

nonn,easy,base

Author

Karl-Heinz Hofmann, Oct 05 2023

Status

approved