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Search: id:A120427
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| A120427 |
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For each y >= 1 there are only finitely many values of x >= 1 such that x-y and x+y are both squares; list all such pairs (x,y) with gcd(x,y) = 1 ordered by values of y; sequence gives y values. |
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+0
2
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| 4, 8, 12, 12, 16, 20, 20, 24, 24, 28, 28, 32, 36, 36, 40, 40, 44, 44, 48, 48, 52, 52, 56, 56, 60, 60, 60, 60, 64, 68, 68, 72, 72, 76, 76, 80, 80, 84, 84, 84, 84, 88, 88, 92, 92, 96, 96, 100, 100, 104, 104, 108, 108, 112, 112, 116, 116, 120, 120, 120, 120, 124, 124, 128
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Ordered even legs of primitive Pythagorean triangles.
Comment from Stephen Waldman, Jun 12 2007: I wrote an arithmetic program once to find out if and when y 'catches up to' n in A120427 (ordered even legs of primitive Pythagorean triples). It's around 16700. As enumerated by the even - or odd - legs, (not sure about the hypotenuses), the triples are 'denser' than the integers.
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REFERENCES
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Donald D. Spencer, Computers in Number Theory, Computer Science Press, Rockville MD, 1982, pp. 130-131.
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FORMULA
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The solutions are given by x = r^2+2*r*k+2*k^2, y = 2*k*(k+r) with r >= 1, k >= 1, r odd, gcd(r, k) = 1.
a(n)=2*A020887(n)=4*A020888(n).
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EXAMPLE
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Pairs are [5, 4], [17, 8], [13, 12], [37, 12], [65, 16], [29, 20], [101, 20], ... E.g. 5-4=1^2, 5+4=3^2.
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CROSSREFS
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Cf. A060829, A061408, A061409.
Even entries of A024355. Ordered union of A081925 and A081935.
Sequence in context: A009012 A046084 A057099 this_sequence A060830 A080458 A147646
Adjacent sequences: A120424 A120425 A120426 this_sequence A120428 A120429 A120430
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), May 02 2001
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EXTENSIONS
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Corrected by Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 12 200 and by Stephen Waldman (brogine(AT)gmail.com), Jun 09 2007
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