A238454 -id:A238454 - cp's OEIS frontend

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

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

A236564 Difference between 2^(2n-1) and the nearest square.

Original entry on oeis.org

1, -1, -4, 7, -17, 23, -89, 7, 28, 112, 448, 1792, -4417, 5503, 22012, -4633, -18532, -74128, -296512, 296863, 1187452, -1181833, -4727332, 4817239, 19268956, -17830441, -71321764, 94338007, 377352028, -9092137, -36368548, -145474192, -581896768, -2327587072, -9310348288

Offset: 1

Alex Ratushnyak, Feb 23 2014

The distances of the even powers 2^(2n) to their nearest squares are obviously all zero and therefore skipped.

			a(1) = 2^1 - 1^2 = 1.
a(2) = 2^3 - 3^2 = -1.
a(3) = 2^5 - 6^2 = 32 - 36 = -4.

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Cf. A053188, A201125, A238454.

A236564 := proc(n)
    local x,sq,lo,hi ;
    x := 2^(2*n-1) ;
    sq := isqrt(x) ;
    lo := sq^2 ;
    hi := (sq+1)^2 ;
    if abs(x-lo) < abs(x-hi) then
        x-lo ;
    else
        x-hi ;
    end if;
end proc: # R. J. Mathar, Mar 13 2014

Table[2^n - Round[Sqrt[2^n]]^2, {n, 1, 79, 2}] (* Alonso del Arte, Feb 23 2014 *)

def isqrt(a):
    sr = 1 << (int.bit_length(int(a)) >> 1)
    while a < sr*sr:  sr>>=1
    b = sr>>1
    while b:
        s = sr + b
        if a >= s*s:  sr = s
        b>>=1
    return sr
for n in range(47):
    nn = 2**(2*n+1)
    a = isqrt(nn)
    d1 = nn - a*a
    d2 = (a+1)**2 - nn
    if d2 < d1:  d1 = -d2
    print(str(d1), end=',')

If A201125(n) < A238454(n), a(n) = A201125(n), otherwise a(n) = -A238454(n). [Negative terms are for cases where the nearest square is above 2^(2n-1), not below it.] - Antti Karttunen, Feb 27 2014

A267032 Difference between smallest integer square >= 10^(2n+1) and 10^(2n+1).

Original entry on oeis.org

6, 24, 489, 4569, 14129, 147984, 2149284, 25191729, 621806289, 5259630921, 19998666404, 102500044289, 3925449108561, 13071591635856, 42248099518244, 4224809951824400, 43007675962234436, 506034404021388356, 6997839444766224, 699783944476622400

Offset: 0

Gwillim Law, Jan 09 2016

nonn

			a(0) = 6 = 4^2 - 10; a(1) = 24 = 32^2 - 1000.

Robert Israel, Table of n, a(n) for n = 0..998
Gwillim Law, blog post, Dec. 12, 2015

Cf. A048761, A068527.

Cf. A238454 (a similar sequence with powers of 2). - Michel Marcus, Jan 17 2016

f:= proc(n) local s;
  s:= isqrt(10^(2*n+1));
  if s^2 < 10^(2*n+1) then s:= s+1 fi;
  s^2 - 10^(2*n+1)
end proc:
seq(f(n),n=0..40); # Robert Israel, Jan 17 2016

dsis[n_]:=Module[{c=10^(2n+1)},(Floor[Sqrt[c]]+1)^2-c]; Array[dsis,20,0] (* Harvey P. Dale, Apr 27 2019 *)

from math import isqrt
def A267032(n): return (isqrt(m:=10**((n<<1)+1))+1)**2-m # Chai Wah Wu, Apr 27 2023

a(n) = A068527(A013715(n)). - Michel Marcus, Jan 17 2016

A362564 a(n) is the largest integer x such that n + 2^x is a square, or -1 if no such number exists.

Original entry on oeis.org

3, 1, 0, 5, 2, -1, 1, 3, 4, -1, -1, 2, -1, 1, 0, 7, 9, -1, -1, 4, 2, -1, 1, 0, -1, -1, -1, 3, -1, -1, -1, 5, 8, 1, 0, 6, -1, -1, -1, -1, 7, -1, -1, -1, 2, -1, 1, 4, 5, -1, -1, -1, -1, -1, -1, 3, 6, -1, -1, 2, -1, 1, 0, 9, 10, -1, -1, 11, -1, -1, -1, -1, 3, -1, -1, -1, 2, -1, 1, 6, -1, -1, -1, 4, -1

Offset: 1

Yifan Xie, Apr 24 2023

sign

a(n) is the maximum integer solution x of the indefinite equation n + 2^x = r^2 where n is a constant and r is a positive integer, or -1 if there are no solutions.
See A247763 for the number of solutions and for further information, references and links about this problem.
If n == 0 (mod 4), we first try x = 0 or 1; when x >= 2, the result can be derived from the result for n/4 (see formula).
If n == 2 (mod 4), the only possible values of x is 1, as otherwise n + 2^x == 2 (mod 4), so nonsquare.
If n == 3 (mod 4), the only possible values of x are 0 and 1, as otherwise n + 2^x == 3 (mod 4), so nonsquare.
If n == 1 (mod 4), or n = 4*k + 1 (k >= 0): we suggest that r = 2*m + 1, 4*k + 1 + 2^x = (2*m + 1)^2, thus k + 2^(x - 2) = m*(m + 1); if k is odd, the only possible values of x is 2 because m*(m + 1) is even.
If n = k^2 (k >= 1), 2^x = (r + k)*(r - k), so 2k must be in the form 2^i - 2^j.
The problem can be solved via reduction to three Mordell curves: n + z^3 = y^2, n + 2z^3 = y^2 or equivalently 4n + (2z)^3 = (2y)^2, n + 4z^3 = y^2 or equivalently 16n + (4z)^3 = (4y)^2, where z := 2^floor(x/3). For a given n, each of these three curves is known to have only a finite number of integer points (y,z), proving that x cannot be unbounded. - Max Alekseyev, Apr 26 2023

			See COMMENTS section for further proof.
For n = 1, 1 + 2^3 = 9 = 3^2;
for n = 4, 4 + 2^5 = 36 = 6^2;
for n = 7, 7 + 2^1 = 9 = 3^2;
for n = 9, 9 + 2^4 = 25 = 5^2.

Thomas Scheuerle, Table of n, a(n) for n = 1..1000

Cf. A000079, A000290, A051204, A234000, A238454, A247763.

def a362564(n): return max((3*v-2*k for k in range(3) for z,, in EllipticCurve([0,4^k*n]).integral_points() if z>=1<Max Alekseyev, Apr 26 2023

a(4*k + 2) = 1 if k + 1 is a square, or -1 otherwise.
a(4*k + 3) = 1 if 4*k + 5 is a square, or 0 is k + 1 is a square and 4*k + 5 is a nonsquare, or -1 otherwise.
a(4*k + 4) = a(k) + 2 if a(k) >= 0, or 0 if 4*k + 1 is a square and a(k) = -1, or -1 otherwise.
a(8*k + 5) = 2 if 8*k + 9 is a square, or -1 otherwise.
a((2^i - 2^j)^2) = i + j + 2 for i,j >= 0.
a(n) > -1 if A247763(n) > 0, or equivalently n is in A051204. - Thomas Scheuerle, May 02 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Formula

A236564 Difference between 2^(2n-1) and the nearest square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A267032 Difference between smallest integer square >= 10^(2*n+1) and 10^(2*n+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A362564 a(n) is the largest integer x such that n + 2^x is a square, or -1 if no such number exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Formula

A267032 Difference between smallest integer square >= 10^(2n+1) and 10^(2n+1).