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A051213
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Numbers of the form 2^x-y^2 >= 0.
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6
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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, 103, 112, 119, 124, 127, 128, 135, 151, 156, 175, 183, 188, 192, 199, 207, 220, 223, 231, 240, 247, 252, 255, 256, 271, 284, 287, 295, 316, 343, 348, 367, 368, 375, 391, 399, 412, 431, 448
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OFFSET
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1,3
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COMMENTS
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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
519 is not in the sequence. [Proof: Consider 2^x-519=y^2 and both sides modulo 3.
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.
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).
The next smaller perfect square is (y-1)^2 = (2^(x/2)-1)^2 = 2^x-2^(1+x/2)+1 .
This must be >=2^x-519 to have a solution, so -2^(1+x/2)+1 >= -519
implies 2^(1+x/2)-1 <= 519, which implies 1+x/2 <= 9.02 and x<=16.
One can check numerically that the range 0<=x<=16 do not form perfect squares 2^x-519.] (End)
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LINKS
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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