A365930 -id:A365930 - cp's OEIS frontend

A365931 a(n) = number of pairs {x,y} with (x,y > 1) such that x^y (= terms of A072103) has bit length <= n.

0, 0, 1, 3, 7, 10, 18, 25, 35, 50, 69, 94, 132, 178, 244, 334, 460, 629, 869, 1201, 1668, 2314, 3223, 4493, 6280, 8793, 12322, 17288, 24286, 34139, 48036, 67630, 95274, 134285, 189349, 267090, 376880, 531942, 750991, 1060463, 1497741, 2115669, 2988957, 4223225, 5967822, 8433889

Offset: 1

Karl-Heinz Hofmann, Oct 07 2023

Number of pairs {x,y} with (x,y > 1) for which x^y < 2^n-1.
In some special cases different pairs have the same result (see A072103 and the example here) and those multiple representations are counted separately.
There is no need to include 2^n-1 because it is a Mersenne number and it cannot be a power anyway.
Limit_{n->oo} a(n)/a(n-1) = sqrt(2) = A002193.
Partial sums of A365930.

			For n = 6: the Mersenne number 2^6-1 = 63 is the largest number with bit length 6 and the upper bound for the following a(6) = 10 powers: 2^2, 2^3, 2^4, 2^5, 3^2, 3^3, 4^2, 5^2, 6^2, 7^2.

Cf. A072103, A002193, A365930 (first differences).

Cf. A017912 (squares), A017981 (cubes).

a[n_] := Sum[Ceiling[2^(n/k)] - 2, {k, 2, n}]; Array[a, 47]

from sympy import integer_nthroot, integer_log
def A365931(n):
    result, nMersenne, new = 0, (1<

a(n) = Sum_{y = 2..n} (ceiling(2^(n/y)) - 2)
a(n) = Sum_{y = 2..n} (floor((2^n-1)^(1/y)) - 1)
a(n) = Sum_{k = 1..n} A365930(k).

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

Original entry on oeis.org

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

Karl-Heinz Hofmann, Oct 05 2023

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.

			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.

Cf. A004632.
Cf. A017981 (partial sums).
Cf. A002580, A126726, A365930.

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

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)])

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.

cp's OEIS Frontend

A365931 a(n) = number of pairs {x,y} with (x,y > 1) such that x^y (= terms of A072103) has bit length <= n.

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