A353140 Digitally balanced numbers (A031443) whose squares and cubes are also digitally balanced.

3274, 13453, 13492, 13706, 14726, 15113, 15498, 15528, 52049, 52251, 52330, 52673, 52778, 53478, 53684, 53775, 53972, 54295, 54411, 54598, 54601, 55057, 55449, 55462, 55505, 55512, 55689, 56333, 58066, 58260, 58446, 58453, 58470, 58918, 59266, 59722, 59786

Offset

1,1

Comments

Numbers x such that x, x^2 and x^3 are terms of A031443, that is, have the same number of 0's as 1's in their binary representations.

All terms are between 2^(2k-1/3) and 2^(2k) for some k. - Robert Israel, Nov 26 2025

Links

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

Maple

isdb:= proc(n) local L; L:= convert(n, base, 2); convert(L, `+`) = nops(L)/2 end proc:
select(t -> isdb(t) and isdb(t^2) and isdb(t^3), [seq($ ceil(2^(2*k-1/3)) .. 2^(2*k)-1, k=1..10)]); # Robert Israel, Nov 26 2025

Mathematica

balQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ @ (m = Length @ d) && Count[d, 1] == m/2]; Select[Range[60000], balQ[#] && balQ[#^2] && balQ[#^3] &] (* Amiram Eldar, Apr 26 2022 *)

Prog

(Python)
from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def isbalanced(n): b = bin(n)[2:]; return b.count("0") == b.count("1")
def A031443gen(): yield from (int("1"+"".join(p), 2) for n in count(1) for p in multiset_permutations("0"*n+"1"*(n-1)))
def agen():
    for k in A031443gen():
        if isbalanced(k**2) and isbalanced(k**3):
            yield k
print(list(islice(agen(), 40))) # Michael S. Branicky, Apr 26 2022

Crossrefs

Cf. A031443, A345397, A353139.

Sequence in context: A184204 A031820 A048959 * A237085 A260410 A223430

Adjacent sequences: A353137 A353138 A353139 * A353141 A353142 A353143

Keyword

nonn,base

Author

Alex Ratushnyak, Apr 26 2022

Status

approved