A164344 Positive integers whose square contains the same number of 0's as 1's when represented in binary.

3, 7, 13, 15, 25, 29, 31, 54, 57, 61, 63, 103, 110, 113, 118, 121, 125, 127, 199, 203, 207, 212, 213, 214, 218, 230, 238, 241, 246, 249, 253, 255, 389, 393, 394, 395, 402, 404, 409, 421, 431, 433, 435, 439, 458, 468, 478, 481, 486, 494, 497, 502, 505, 509, 511

Offset: 1

Author

Leroy Quet, Aug 13 2009

Keywords

• base
• nonn

Comments

The squares must have an even number of binary digits, given by ceiling(log_2(a(n)^2)) = ceiling(2 log_2 a(n)), or equivalently, 2^(k-1/2) < a(n) < 2^k for some integer k > 0, which explains the jumps in the graph of the sequence. - M. F. Hasler, Jul 12 2022

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• James Propp et al., Perfectly balanced perfect squares, math-fun mailing list (archive available to subscribers), Jul 12 2022.

Crossrefs

Cf. A031443 (digitally balanced numbers), A164343 (squares of the terms), A000120 (Hamming weight), A070939 (number of binary digits).

Programs

Mathematica

sn01Q[n_]:=Module[{idn2=IntegerDigits[n^2,2]},Count[idn2,1] == Length[ idn2]/2]; Select[Range[600],sn01Q] (* Harvey P. Dale, Apr 03 2016 *)

PARI

select( {is_A164344(n)=hammingweight(n^2)*2==exponent(n^2*2)}, [0..512]) \\ M. F. Hasler, Jul 12 2022

Python

def bal(n): return n and n.bit_length() == n.bit_count() * 2
print([k for k in range(512) if bal(k*k)]) # Michael S. Branicky, Jul 12 2022

Formula

{n | n^2 is in A031443} = {n | 2*A000120(n^2) = A070939(n^2)}, i.e., twice the Hamming weight must equal the number of binary digits, for the squares of the terms. - M. F. Hasler, Jul 12 2022

Extensions

More terms from Sean A. Irvine, Oct 08 2009