A164344 -id:A164344 - cp's OEIS frontend

A164343 A positive integer is included if it is a square that contains the same number of 0's as 1's when represented in binary.

9, 49, 169, 225, 625, 841, 961, 2916, 3249, 3721, 3969, 10609, 12100, 12769, 13924, 14641, 15625, 16129, 39601, 41209, 42849, 44944, 45369, 45796, 47524, 52900, 56644, 58081, 60516, 62001, 64009, 65025, 151321, 154449, 155236, 156025, 161604, 163216, 167281

Offset: 1

Leroy Quet, Aug 13 2009

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A031443, A164344.

Select[Range[500]^2,DigitCount[#,2,1]==DigitCount[#,2,0]&] (* Harvey P. Dale, Nov 18 2014 *)

def bal(n): return n and n.bit_length() == n.bit_count() * 2
print([s for s in (k*k for k in range(403)) if bal(s)]) # Michael S. Branicky, 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

cp's OEIS Frontend

A164343 A positive integer is included if it is a square that 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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Python