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Search: id:A111205
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| A111205 |
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Square-free numbers n such that the difference between the closest squares surrounding n and n have a common divisor greater than 1. |
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+0
1
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| 14, 18, 21, 24, 33, 39, 50, 51, 54, 55, 57, 60, 63, 68, 95, 102, 105, 108, 111, 112, 114, 117, 119, 120, 138, 145, 150, 155, 160, 165, 171, 174, 177, 180, 183, 186, 189, 192, 195, 203, 248, 258, 261, 264, 267, 270, 273, 275, 276, 279, 282, 285, 286, 288, 290
(list; graph; listen)
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OFFSET
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6,1
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COMMENT
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Conjecture: The number of terms in this sequence is infinite.
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FORMULA
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Let n be a squarefree composite number and r = floor(sqrt(n)). Then the closest surrounding squares of n are r^2 and (r+1)^2. So d = (r+1)^2 - r^2 = 2r+1. If gcd(n, d) > 1 then list n.
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EXAMPLE
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14 is a squarefree composite number. 3^2 and 4^2 are the closest squares
surrounding 14. So the difference, 16-9 = 7 and 14 have a common divisor
greater than 1 namely 7, so 14 is the first entry in the table.
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PROGRAM
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(PARI) surrsqgcd(n) = { local(x, y, j, r, d); for(x=1, n, if(!issquare(x)&!isprime(x), r=floor(sqrt(x)); d=r+r+1; if(gcd(x, d) > 1, print1(x", ") ) )) }
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CROSSREFS
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Sequence in context: A060504 A052026 A118499 this_sequence A097324 A051419 A000053
Adjacent sequences: A111202 A111203 A111204 this_sequence A111206 A111207 A111208
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KEYWORD
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easy,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Nov 12 2005
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