A353139 Digitally balanced numbers (A031443) whose squares are also digitally balanced.

212, 781, 794, 806, 817, 838, 841, 844, 865, 2962, 3101, 3130, 3171, 3178, 3185, 3213, 3219, 3226, 3269, 3274, 3335, 3353, 3354, 3356, 3370, 3378, 3490, 3496, 3521, 3528, 3595, 3597, 3606, 3610, 3626, 3651, 3672, 3718, 3777, 11797, 11798, 11850, 11938, 12049

Offset: 1

Author

Alex Ratushnyak, Apr 26 2022

Keywords

• nonn
• base

Comments

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

Crossrefs

Cf. A031443, A345397.

Programs

Mathematica

balQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ @ (m = Length @ d) && Count[d, 1] == m/2]; Select[Range[12000], balQ[#] && balQ[#^2] &] (* Amiram Eldar, Apr 26 2022 *)

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):
            yield k
print(list(islice(agen(), 40))) # Michael S. Branicky, Apr 26 2022