A356878 -id:A356878 - cp's OEIS frontend

A357658 a(n) is the maximum Hamming weight of squares k^2 in the range 2^n <= k^2 < 2^(n+1).

1, 2, 3, 3, 5, 4, 6, 6, 8, 8, 9, 9, 13, 11, 13, 12, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 30, 31, 31, 34, 33, 34, 37, 37, 38, 38, 39, 39, 41, 41, 42, 44, 44, 44, 46, 47, 47, 49, 50, 51, 52, 52, 53, 54, 55, 55, 57, 57, 58, 59, 62, 63

Offset: 2

Hugo Pfoertner, Oct 09 2022

The sequence can be approximated by a linear function c*n + d, with c ~= 0.883 +- 0.003, d ~= -1.65 +- 0.16. See linked plot. For a square number with 100 binary digits (n=99) a maximum Hamming weight of 85 or 86 is expected. For example, 1125891114428899^2 has Hamming weight 85.

			  n         A357753(n) a(n) A357659(n)    A357660(n)    A357754(n)
  bits  2^n  least sq  Ha w  k_min  ^2     k_max  ^2   largest sq
   2     4      4       1     2      4      2      4        4
   3     8      9       2     3      9      3      9        9
   4    16     16       3     5     25      5     25       25
   5    32     36       3     7     49      7     49       49
   6    64     64       5    11    121     11    121      121
   7   128    144       4    13    169     15    225      225
  12  4096   4096       9    75   5625     89   7921     8100

Hugo Pfoertner, Table of n, a(n) for n = 2..125, including results from Bert Dobbelaere and users l4m2, gsitcia, anttiP in Code Golf challenge (terms 103..125)
Code Golf Stackexchange, Smallest and largest 100-bit square with maximum Hamming weight, fastest code challenge started Dec 15 2022.
Hugo Pfoertner, Maximum of bits=1 in squares 2^n < k^2 < 2^(n+1), plot of linear fit (December 2022).

Cf. A000120, A000290, A356878, A357304, A357753, A357754.

A357659 and A357660 are the minimal and the maximal values of k producing a(n).

A357656 a(n) is a lower bound for the largest Hamming weight of squares with exactly n binary zeros.

Original entry on oeis.org

1, 0, 13, 8, 13, 16, 37, 38, 44

Offset: 0

Karl-Heinz Hofmann and Hugo Pfoertner, Oct 07 2022

The terms from a(2) onwards must be regarded as lower bounds, because no proof for the non-existence of squares with a very small number of binary zeros in the range k^2 > 2^90 (see b-file of A230097) is known.

a(9) >= 63, a(10) >= 57.

			                A357657(n)
   n  a(n) bits     k       k^2          k^2 in binary
   0    1    1      1         1                      1
   1    0    1      0         0                      0
   2   13   15    181     32761        111111111111001
   3    8   11     45      2025            11111101001
   4   13   17    362    131044      11111111111100100
   5   16   21   1241   1540081  101110111111111110001

A357657 are the corresponding square roots of the record-setting squares.

Cf. A000120, A000290, A159918, A230097, A356878, A357304, A357305.

cp's OEIS Frontend

A357658 a(n) is the maximum Hamming weight of squares k^2 in the range 2^n <= k^2 < 2^(n+1).

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