A381720 Integers whose Hamming weight is a cube.

0, 1, 2, 4, 8, 16, 32, 64, 128, 255, 256, 383, 447, 479, 495, 503, 507, 509, 510, 512, 639, 703, 735, 751, 759, 763, 765, 766, 831, 863, 879, 887, 891, 893, 894, 927, 943, 951, 955, 957, 958, 975, 983, 987, 989, 990, 999, 1003, 1005, 1006, 1011, 1013, 1014

Offset: 1

Author

Ctibor O. Zizka, Mar 05 2025

Keywords

• nonn
• base

Comments

The plot of terms of the form k - 2^floor(log_2(k)) shows quasi-periodic structures on the intervals [2^i, 2^(i+1)]. In general, all sequences of the form "Integers whose Hamming weight is f(x)", where f(x) is an integer valued function, are quasi-periodic on intervals [2^i, 2^(i+1)].

The powers of 2 (A000079) are terms.

A023690 is a subsequence.

Examples

			For k = 255: A000120(255) = 8 = 2^3 is a cube, thus 255 is a term.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000079, A000120, A000578, A023690.

Cf. A084561, A344602.

Programs

Maple

filter:= proc(n) type(surd(convert(convert(n,base,2),`+`),3),integer) end proc:
select(filter, [$0..2000]); # Robert Israel, May 18 2025

Mathematica

Select[Range[0, 1200], IntegerQ[Surd[DigitCount[#, 2, 1], 3]] &] (* Amiram Eldar, Mar 05 2025 *)

PARI

isok(k) = ispower(hammingweight(k), 3); \\ Michel Marcus, Mar 05 2025

Python

from itertools import count, islice, combinations
from sympy import integer_nthroot
def A381720_gen(): # generator of terms
    a = []
    yield 0
    for l in count(1):
        b = 1<A381720_list = list(islice(A381720_gen(),53)) # Chai Wah Wu, Mar 06 2025