A357304 -id:A357304 - cp's OEIS frontend

A230097 Indices of records in A159918.

0, 1, 3, 5, 11, 21, 39, 45, 75, 155, 181, 627, 923, 1241, 2505, 3915, 5221, 6475, 11309, 15595, 19637, 31595, 44491, 69451, 113447, 185269, 244661, 357081, 453677, 908091, 980853, 2960011, 2965685, 5931189, 11862197, 20437147, 22193965, 43586515, 57804981, 157355851

Offset: 1

N. J. A. Sloane, Oct 11 2013

The records themselves are not so interesting: 0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, ... (A357304).

Lindström mentions that the record value 34 in A159918 is first reached at n = 980853.

Bert Dobbelaere, Table of n, a(n) for n = 1..80, (terms 41..64 from Donovan Johnson, 65..70 from Hugo Pfoertner, missing 68 and 72..80 from Bert Dobbelaere).
Bernt Lindström, On the binary digits of a power, Journal of Number Theory, Volume 65, Issue 2, August 1997, Pages 321-324.

Cf. A000120, A159918, A231897, A357304, A357658.

a230097 n = a230097_list !! (n-1)
a230097_list = 0 : f 0 0 where
   f i m = if v > m then i : f (i + 1) v else f (i + 1) m
           where v = a159918 i
-- Reinhard Zumkeller, Oct 12 2013
(Python 3.10+)
from itertools import count, islice
def A230097_gen(): # generator of terms
    c = -1
    for n in count(0):
        if (m := (n**2).bit_count())>c:
            yield n
            c = m
A230097_list = list(islice(A230097_gen(),20)) # Chai Wah Wu, Oct 01 2022

Lindström shows that lim sup wt(m^2)/log_2 m = 2.

a(19)-a(40) from Reinhard Zumkeller, Oct 12 2013

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

Original entry on oeis.org

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.

A357657 a(n) is a lower bound for the square root of the maximum square with exactly n zeros in its binary representation.

Original entry on oeis.org

1, 0, 181, 45, 362, 1241, 2965685, 5931189, 57804981

Offset: 0

Karl-Heinz Hofmann and Hugo Pfoertner, Oct 08 2022

See A357656 for more information.

a(9) >= 66537313397, a(10) >= 10520476455.

A357656 gives the Hamming weight of the squared terms.

Cf. A000120, A000290, A159918, A230097, A356838, A357304, A357305, A357753, A357754.

A357742 a(n) is the maximum binary weight of the squares of n-bit numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 13, 13, 15, 16, 18, 20, 22, 24, 25, 27, 29, 31, 34, 34, 37, 38, 39, 41, 44, 44, 47, 49, 51, 52, 54, 55, 57, 59, 63, 63, 64, 66, 68, 69, 72, 73, 76, 77, 78, 80, 82, 85, 87

Offset: 1

Karl-Heinz Hofmann and Hugo Pfoertner , Oct 11 2022

			   bit   |
  length |          possible binary weight of k^2
   of k  | 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
   = n   |          the rightmost value is a(n)
  -------+--------------------------------------------------------------
     1   | 0  1
     2   |    1  2  -  -
     3   |    1  2  3  -  -  -
     4   |    1  2  3  4  5  -  -  -
     5   |    1  2  3  4  5  6  -  -  -  -
     6   |    1  2  3  4  5  6  7  8  -  -  -  -
     7   |    1  2  3  4  5  6  7  8  9  -  -  -  -  -
     8   |    1  2  3  4  5  6  7  8  9 10 11  - 13  -  -  -
     9   |    1  2  3  4  5  6  7  8  9 10 11 12 13  -  -  -  -  -
    10   |    1  2  3  4  5  6  7  8  9 10 11 12 13 14 15  -  -  -  -  -

Cf. A000290, A159918, A357304, A357658.

a(n) = max(A357658(2*n-2), A357658(2*n-1)).

a(47)-a(50) from Martin Ehrenstein, Dec 26 2023

cp's OEIS Frontend

A230097 Indices of records in A159918.

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

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Author

Keywords

Comments

Examples

Crossrefs

A357657 a(n) is a lower bound for the square root of the maximum square with exactly n zeros in its binary representation.

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Author

Keywords

Comments

Crossrefs

A357742 a(n) is the maximum binary weight of the squares of n-bit numbers.

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Author

Keywords

Examples

Crossrefs

Formula

Extensions