A357754 -id:A357754 - 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).

A357753 a(n) is the least square with n binary digits.

Original entry on oeis.org

4, 9, 16, 36, 64, 144, 256, 529, 1024, 2116, 4096, 8281, 16384, 33124, 65536, 131769, 262144, 525625, 1048576, 2099601, 4194304, 8392609, 16777216, 33558849, 67108864, 134235396, 268435456, 536895241, 1073741824, 2147488281, 4294967296, 8589953124, 17179869184

Offset: 3

Hugo Pfoertner, Oct 11 2022

Cf. A000290, A000302, A065732, A070939, A017912, A357754.

a:= n-> ceil(sqrt(2)^(n-1))^2:
seq(a(n), n=3..35);  # Alois P. Heinz, Oct 13 2022

Array[Ceiling[Sqrt[2^(# - 1)]]^2 &, 33, 3]

for (n=3, 35, for(k=0, oo, if(#digits(k^2,2)==n, print1(k^2,", "); break)))

a(n) = if(n%2 == 1, 1 << (n-1), ceil(sqrt(1<<(n-1)))^2) \\ David A. Corneth, Oct 11 2022

from math import isqrt
def A357753(n): return 1<Chai Wah Wu, Oct 13 2022

a(n) = A017912(n-1)^2.

a(2n+1) = (2^n)^2 = 4^n, for n>=1; indeed: 4^n_{10} = 10^(2n){2} that is the least number with 2n+1 binary digits. - _Bernard Schott, Oct 15 2022

A056007 Difference between 2^n and largest square strictly less than 2^n.

Original entry on oeis.org

1, 1, 3, 4, 7, 7, 15, 7, 31, 28, 63, 23, 127, 92, 255, 7, 511, 28, 1023, 112, 2047, 448, 4095, 1792, 8191, 7168, 16383, 5503, 32767, 22012, 65535, 88048, 131071, 166831, 262143, 296599, 524287, 444943, 1048575, 296863, 2097151, 1187452, 4194303

Offset: 0

Henry Bottomley, Jul 24 2000

Note that this is not a strictly ascending sequence. - Alonso del Arte, Apr 28 2022

			a(5) = 2^5 - 5^2 =  7;
a(6) = 2^6 - 7^2 = 15.

Cf. A000079, A017912, A071797, A357754.

Cf. A051213, A056008.

Table[2^n - Floor[Sqrt[2^n - Boole[EvenQ[n]]]]^2, {n, 0, 47}] (* Alonso del Arte, Apr 28 2022 *)

a(n) = if(n%2, sqrtint(1<Kevin Ryde, Oct 12 2022

from math import isqrt
def a(n): return 2**n - isqrt(2**n-1)**2
print([a(n) for n in range(43)]) # Michael S. Branicky, Apr 29 2022

a(n) = 2^n - (ceiling(2^(n/2)) - 1)^2 = A000079(n) - (A017912(n) - 1)^2. - Vladeta Jovovic, May 01 2003
a(n) = A071797(A000079(n)). - Michel Marcus, Apr 29 2022
a(n) = 2^n - A357754(n). - Kevin Ryde, Oct 12 2022

A116601 a(0) = a(1) = 0; for n >= 2, a(n) = floor(sqrt(2^(n-2)-1)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 31, 45, 63, 90, 127, 181, 255, 362, 511, 724, 1023, 1448, 2047, 2896, 4095, 5792, 8191, 11585, 16383, 23170, 32767, 46340, 65535, 92681, 131071, 185363, 262143, 370727, 524287, 741455, 1048575, 1482910, 2097151, 2965820, 4194303, 5931641

Offset: 0

Roger L. Bagula, Mar 28 2006

nonn

Numbers k such that k^2 is the largest square less than the next power of 2. - Hugo Pfoertner, Sep 30 2022

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A002193, A357754.

Join[{0, 0}, Table[Floor[Sqrt[2^(n - 2) - 1]], {n, 2, 50}]] (* G. C. Greubel, Oct 28 2017 *)

from math import isqrt
def A116601(n): return isqrt((1< 1 else 0 # Chai Wah Wu, Oct 13 2022

Edited by N. J. A. Sloane, May 10 2007

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.

A357752 a(n) is the largest perfect power < 2^n.

Original entry on oeis.org

4, 9, 27, 49, 125, 243, 484, 1000, 2025, 3969, 8100, 16129, 32761, 65025, 131044, 261121, 524176, 1046529, 2096704, 4190209, 8386816, 16769025, 33547264, 67092481, 134212225, 268402689, 536848900, 1073676289, 2147395600, 4294836225, 8589767761, 17179607041, 34359441769

Offset: 3

Hugo Pfoertner, Oct 12 2022

nonn

Are all terms > 1000 identical to A357754?

Cf. A000079, A001597, A357751, A357754.

for (n=3, 35, forstep (k=2^n-1, 0, -1, if(ispower(k), print1(k,", "); break)))

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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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A357753 a(n) is the least square with n binary digits.

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A056007 Difference between 2^n and largest square strictly less than 2^n.

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A116601 a(0) = a(1) = 0; for n >= 2, a(n) = floor(sqrt(2^(n-2)-1)).

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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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A357752 a(n) is the largest perfect power < 2^n.

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