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Search: id:A135789
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| A135789 |
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Positive numbers of the form x^4-6x^2 y^2+y^4 (where x,y are integers). |
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+0
9
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| 28, 41, 161, 448, 476, 656, 721, 956, 1081, 1241, 1393, 2108, 2268, 2576, 3281, 3321, 3713, 3836, 4633, 4681, 5593, 6076, 7168, 7616, 8188, 9401, 9641, 10496, 11536, 11753, 12121, 12593, 13041, 13916, 15296, 16828, 17296, 17500, 19516, 19856
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Squares of these numbers are of the form N^4-M^2 (where N belongs to A135786 and M to A057102). Proof is based on the identity (x^4 - 6x^2 y^2 + y^4)^2=(x^2+y^2)^4-(4(x^3y-xy^3))^2
Since x^4-6x^2y^2+y^4=d*d' where d=x^2-y^2+2xy and d'=x^2-y^2-2xy, and d-d'=4xy, the computational technique is to consider the divisors d|n, d'=n/d, to check that the difference is a mutliple of 4, and to check x in the range 1..d/3. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 18 2009]
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MAPLE
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isA135789 := proc(n) for d in numtheory[divisors](n) do dprime := n/d ; if abs(d-dprime) mod 4 = 0 then for x from 1 to d/3 do y := (d-dprime)/4/x ; if type(y, 'integer') and y< x and y> 0 then if n = (x^2-y^2+2*x*y)*(x^2-y^2-2*x*y) then RETURN(true); fi; fi; od: fi: od: RETURN(false) ; end: for n from 1 do if isA135789(n) then printf("%d, \n", n) ; fi; od: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 18 2009]
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MATHEMATICA
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a = {}; Do[Do[w = x^4 - 6x^2 y^2 + y^4; If[w > 0&&w<10000, AppendTo[a, w]], {x, y, 2000}], {y, 1, 2000}]; Union[a]
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CROSSREFS
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Cf. A000404, A050803, A057102, A135784, A060803, A135786, A135787.
Sequence in context: A109798 A084807 A047630 this_sequence A154035 A025368 A025359
Adjacent sequences: A135786 A135787 A135788 this_sequence A135790 A135791 A135792
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KEYWORD
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nonn
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AUTHOR
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Artur Jasinski (grafix(AT)csl.pl), Nov 29 2007, Nov 14 2008
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EXTENSIONS
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More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 18 2009
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