A056007 -id:A056007 - cp's OEIS frontend

A357754 a(n) is the largest square with n binary digits.

4, 9, 25, 49, 121, 225, 484, 961, 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 11 2022

Cf. A000290, A070939, A056007, A116601, A357753.

Array[Floor[Sqrt[2^# - 1]]^2 &, 33, 3] (* Michael De Vlieger, Oct 11 2022 *)

for (n=3, 35, forstep (k2=2^n-1, 2^(n-1), -1, if (issquare(k2), print1(k2,", "); break)))

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

from math import isqrt
def A357754(n): return (isqrt((1<>1))-1)**2 # Chai Wah Wu, Oct 13 2022

a(n) = 2^n - A056007(n). - Kevin Ryde, Oct 13 2022

a(n) = A116601(n+2)^2. - Michel Marcus, Oct 13 2022

A056008 Difference between (smallest square strictly greater than 2^n) and 2^n.

Original entry on oeis.org

3, 2, 5, 1, 9, 4, 17, 16, 33, 17, 65, 68, 129, 89, 257, 356, 513, 697, 1025, 1337, 2049, 2449, 4097, 4001, 8193, 4417, 16385, 17668, 32769, 24329, 65537, 4633, 131073, 18532, 262145, 74128, 524289, 296512, 1048577, 1186048, 2097153, 1778369

Offset: 0

Henry Bottomley, Jul 24 2000

nonn

If n is even, a(n) = 2*2^(n/2) + 1, since 2^n = (2^(n/2))^2, and a(n) = (2^(n/2) + 1)^2 - (2^(n/2))^2 = 2*2^(n/2) + 1. - Jean-Marc Rebert, Mar 02 2016

If n is odd, a(n) = 4*a(n-2) or 4*a(n-2) - 4*sqrt(a(n-2) + 2^(n-2)) + 1. - Robert Israel, Mar 02 2016

			a(5)=6^2-2^5=4; a(6)=9^2-2^6=17

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Bisections: A000051, A238454.
Cf. A000079, A017910.
Cf. A051204, A056007.

[(Floor(2^(n/2))+1)^2-2^n : n in [0..50]]; // Vincenzo Librandi, Mar 03 2016

f:= proc(n) local m;
   if n::even then m:= 2*2^(n/2)+1
   else m:= ceil(sqrt(2)*2^((n-1)/2))
   fi;
   m^2-2^n
end proc:
map(f, [$0..100]); # Robert Israel, Mar 02 2016

ssg[n_]:=Module[{s=2^n},(1+Floor[Sqrt[s]])^2-s]; Array[ssg,50,0] (* Harvey P. Dale, Aug 22 2015 *)
Table[((Floor[2^(n/2)] + 1)^2 - 2^n), {n, 0, 50}] (* Vincenzo Librandi, Mar 03 2016 *)

from math import isqrt
def A056008(n): return (isqrt(m:=1<Chai Wah Wu, Apr 28 2023

a(n) = (floor(2^(n/2))+1)^2 - 2^n = (A017910(n)+1)^2 - A000079(n). - Vladeta Jovovic, May 01 2003

a(2k) = 2*2^k + 1 = 2*a(2(k-1)) - 1. - Jean-Marc Rebert, Mar 02 2016

A248346 Primes of the form 2^x - y^2, with y^2 < 2^x.

Original entry on oeis.org

2, 3, 7, 23, 31, 47, 71, 79, 103, 127, 151, 199, 223, 271, 367, 431, 463, 487, 503, 727, 751, 823, 967, 1087, 1303, 1319, 1423, 1439, 1559, 1607, 1759, 1823, 1879, 1951, 1999, 2039, 2143, 3343, 3527, 3623, 3967, 4447, 4943, 5167, 5503, 5591, 5791, 6199, 6343

Offset: 1

Juri-Stepan Gerasimov, Oct 05 2014

nonn

Primes in A051213.

			7 is in this sequence because 7 = 2^3 - 1^2 = 2^4 - 3^2 = 2^5 - 5^2 = 2^7 - 11^2 = 2^15 - 181^2.
1559 is in this sequence because 1559 = 2^19 - 723^2 is prime. - _Sean A. Irvine_, Apr 28 2022

Cf. A038198, A051213, A060728, A248344.

Primes in A056007 form a subset of the numbers in this sequence.

Select[Union[Flatten[Table[2^x - y^2, {x, 16}, {y, 0, Floor[Sqrt[2^x]]}]]], PrimeQ] (* Alonso del Arte, Oct 05 2014 *)

a(24)-a(38) from Alonso del Arte, Oct 05 2014

More terms and missing terms inserted by Sean A. Irvine, Apr 28 2022

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A357754 a(n) is the largest square with n binary digits.

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A056008 Difference between (smallest square strictly greater than 2^n) and 2^n.

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A248346 Primes of the form 2^x - y^2, with y^2 < 2^x.

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