A357305 -id:A357305 - cp's OEIS frontend

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

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.