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A056901
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Least semi-perimeter s of primitive Pythagorean triangle with inradius n.
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1
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6, 15, 20, 45, 42, 35, 72, 153, 110, 63, 156, 77, 210, 99, 88, 561, 342, 143, 420, 117, 130, 195, 600, 209, 702, 255, 812, 165, 930, 187, 1056, 2145, 238, 399, 204, 221, 1482, 483, 304, 273, 1806, 247, 1980, 285, 266, 675, 2352, 665, 2550, 783, 460, 357
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| For a primitive Pythagorean triangle with sides X, Y & Z, we have two generating numbers m&n such that m>n, gcd(m,n) = 1 and the parity of m&n are opposite. X = m^2 - n^2, Y = 2mn and Z = m^2 + n^2, s = m^2 + mn and finally r = n(m-n).
Moreover, a primitive Pythagorean triangle has area n*a(n).
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REFERENCES
| Albert H. Beiler, "Recreations In The Theory Of Numbers, The Queen Of Mathematics Entertains," Dover Publications, Inc., Second Edition, NY, 1966, Chapter XIV, 'The Eternal Triangle,' pages 104 - 134.
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LINKS
| Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics
Wm. H. Richardson, The inradius of a Right Triangle with Integral Sides
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FORMULA
| When n is i)an odd prime power, s = (n + 1)(n + 2). ii)a power of 2, s = (n + 1)(2n + 1). iii)a composite with relatively prime factors a*b such that a is smallest, s = (a + b)(2a + b).
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MATHEMATICA
| a = Table[10^9, {75} ]; Do[ If[ GCD[m, n] == 1 && Sort[ Mod[ {m, n}, 2]] == {0, 1}, s = m^2 + m*n; r = n(m - n); If[r < 76 && a[[r]] > s, a[[r]] = s; Print[r, " ", s]]], {m, 2, 10^2}, {n, 1, m - 1} ]
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CROSSREFS
| Cf. A014498.
Sequence in context: A094183 A196394 A162693 * A012412 A009092 A015793
Adjacent sequences: A056898 A056899 A056900 * A056902 A056903 A056904
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KEYWORD
| nonn
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AUTHOR
| Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 12 2002
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EXTENSIONS
| Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 18 2002
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