A051213 - OEIS

%I #30 May 13 2017 23:30:06

%S 0,1,2,3,4,7,8,12,15,16,23,28,31,32,39,47,48,55,60,63,64,71,79,87,92,

%T 103,112,119,124,127,128,135,151,156,175,183,188,192,199,207,220,223,

%U 231,240,247,252,255,256,271,284,287,295,316,343,348,367,368,375,391,399,412,431,448

%N Numbers of the form 2^x-y^2 >= 0.

%C Is 519 in this sequence? Then this is the value of a(73), else it is 527, after which the sequence goes on with 540, 583, 604, 615, 623, 624,... - _M. F. Hasler_, Oct 09 2014

%C From _R. J. Mathar_, Oct 21 2014: (Start)

%C 519 is not in the sequence. [Proof: Consider 2^x-519=y^2 and both sides modulo 3.

%C Then 2^x-519 = 1,2,1,2.... (mod 3) for x>=0 and y^2=0,1,1,0,1,1,... (mod 3) for y>=0.

%C For moduli to match (i.e, both 1), x must be even. Then 2^x is the square of the integer y=2^(x/2). (Note that this reference does not work in integers if x is odd).

%C The next smaller perfect square is (y-1)^2 = (2^(x/2)-1)^2 = 2^x-2^(1+x/2)+1 .

%C This must be >=2^x-519 to have a solution, so -2^(1+x/2)+1 >= -519

%C implies 2^(1+x/2)-1 <= 519, which implies 1+x/2 <= 9.02 and x<=16.

%C One can check numerically that the range 0<=x<=16 do not form perfect squares 2^x-519.] (End)

%H M. F. Hasler, <a href="/A051213/b051213.txt">Table of n, a(n) for n = 1..72</a>

%H J. Cohn, <a href="https://eudml.org/doc/206586">The diophantine equation x^2+C=y^n</a>, Acta Arithm. 65 (4) (1993) 367-381

%H Fadwa S. Abu Muriefah, Yann Bugeaud, <a href="https://eudml.org/doc/232379">The diophantine equation x^2+c=y^n: a brief overview</a>, Rev. Colomb. Matem. 40 (1) (2006) 31-37

%t max = 1000; Clear[f]; f[m_] := f[m] = Select[Table[2^x - y^2, {x, 0, m}, {y, 0, Ceiling[2^(x/2)]}] // Flatten // Union, 0 <= # <= max &]; f[1]; f[m = 2]; While[f[m] != f[m - 1], m++]; Print["m = ", m]; A051213 = f[m] (* _Jean-François Alcover_, May 13 2017 *)

%o (PARI) is_A051213(n)=!A200522(n) \\ _M. F. Hasler_, Oct 09 2014

%Y Cf. A201125.

%K nonn

%O 1,3

%A _David W. Wilson_

%E More terms from _M. F. Hasler_, Oct 09 2014