A164343 -id:A164343 - cp's OEIS frontend

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

Leroy Quet, Aug 13 2009

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

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.

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

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

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

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

{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

More terms from Sean A. Irvine, Oct 08 2009

A356878 a(n) is the least number of binary zeros of squares with binary weight n.

Original entry on oeis.org

1, 0, 2, 2, 4, 2, 3, 4, 3, 4, 5, 5, 5, 2, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 7, 6, 8, 9, 6, 7, 8, 9, 8, 9, 9, 8, 10, 9, 9, 10, 9, 9, 9, 9, 10, 10, 10, 11, 10

Offset: 0

Karl-Heinz Hofmann, Sep 30 2022

			    Squares with    |             possible number of zeros
    binary weight   |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
         n          |            the leftmost value is a(n)
   -----------------+----------------------------------------------------
         0          |  -  1  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
         1          |  0  -  2  -  4  -  6  -  8  - 10  - 12  - 14  - ...
         2          |  -  -  2  -  4  -  6  -  8  - 10  - 12  - 14  - ...
         3          |  -  -  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
         4          |  -  -  -  -  4  -  6  -  8  - 10  - 12  - 14  - ...
         5          |  -  -  2  -  4  5  6  7  8  9 10 11 12 13 14 15 ...
         6          |  -  -  -  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
         7          |  -  -  -  -  4  5  6  7  8  9 10 11 12 13 14 15 ...
         8          |  -  -  -  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
         9          |  -  -  -  -  4  5  6  7  8  9 10 11 12 13 14 15 ...
        10          |  -  -  -  -  -  5  6  7  8  9 10 11 12 13 14 15 ...
        11          |  -  -  -  -  -  5  6  7  8  9 10 11 12 13 14 15 ...
        12          |  -  -  -  -  -  5  6  7  8  9 10 11 12 13 14 15 ...
        13          |  -  -  2  -  4  -  6  7  8  9 10 11 12 13 14 15 ...
        14          |  -  -  -  -  -  5  6  7  8  9 10 11 12 13 14 15 ...
         :          |  ...

Karl-Heinz Hofmann, Table (like in example) with n = 0..57

Cf. A000120, A164343, A164344.

A357750 a(n) is the least k such that B(k^2) - B(k) = n, where B(m) is the binary weight A000120(m).

Original entry on oeis.org

0, 5, 11, 21, 45, 75, 217, 331, 181, 789, 1241, 2505, 5701, 5221, 11309, 19637, 43151, 69451, 82709, 166027, 346389, 607307, 689685, 1458357, 1380917, 2507541, 5906699, 2965685, 5931189, 11862197, 47448787, 82188309, 57804981, 94905541, 188883211, 373457573, 640164021

Offset: 0

Karl-Heinz Hofmann and Hugo Pfoertner, Oct 17 2022

			  ----------------------------------------------------
  n     k      k^2     binary k             binary k^2
  ----------------------------------------------------
  0     0        0            0                      0
  1     5       25          101                  11001
  2    11      121         1011                1111001
  3    21      441        10101              110111001
  4    45     2025       101101            11111101001
  5    75     5625      1001011          1010111111001
  6   217    47089     11011001       1011011111110001
  7   331   109561    101001011      11010101111111001
  8   181    32761     10110101        111111111111001
  9   789   622521   1100010101   10010111111110111001

Cf. A000120, A000290, A159918, A164343, A164344, A356877, A357658.

a(n) = my(k=0); while(hammingweight(k^2) - hammingweight(k) != n, k++); k;

from itertools import count
def A357750(n):
    for k in count(0):
        if (k**2).bit_count()-k.bit_count()==n:
            return k # Chai Wah Wu, Oct 17 2022

A345348 Triangular numbers that in base 2 have the same number of 0's and 1's.

Original entry on oeis.org

10, 153, 210, 595, 666, 820, 2278, 2701, 9045, 9870, 10585, 11476, 12403, 13366, 13861, 14365, 34191, 34716, 35245, 36046, 37675, 37950, 39340, 39621, 40470, 41905, 42195, 42778, 43365, 44551, 45150, 45451, 46665, 48516, 49455, 50086, 50403, 51681, 52003, 52326

Offset: 1

Ctibor O. Zizka, Jun 15 2021

			Triangular number 153 = '10011001' in binary, the number of 1's equals the number of 0's, so 153 is a term.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A000217 and A031443.

Cf. A164343.

Select[Table[n*(n + 1)/2, {n, 0, 330}], Equal @@ DigitCount[#, 2] &] (* Amiram Eldar, Jun 15 2021 *)

isA031443(n)=2*hammingweight(n)==exponent(n)+1
list(lim)=my(v=List(),n=4,t); while((t=n*n++/2)<=lim, if(isA031443(t), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Jun 21 2021

A345348_list = [n for n in (m*(m+1)//2 for m in range(10**6)) if len(bin(n))-2 == 2*bin(n).count('1')] # Chai Wah Wu, Jun 21 2021

More terms from Jinyuan Wang, Jun 15 2021

cp's OEIS Frontend

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

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A356878 a(n) is the least number of binary zeros of squares with binary weight n.

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A357750 a(n) is the least k such that B(k^2) - B(k) = n, where B(m) is the binary weight A000120(m).

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A345348 Triangular numbers that in base 2 have the same number of 0's and 1's.

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