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A056901
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Least semiperimeter 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
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OFFSET
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1,1
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COMMENTS
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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
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Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.
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
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Table of n, a(n) for n=1..52.
Eric Weisstein's World of Mathematics, Semiperimeter
Wm. H. Richardson, The inradius of a Right Triangle with Integral Sides
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FORMULA
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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
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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
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Cf. A014498.
Sequence in context: A094183 A196394 A162693 * A208542 A012412 A009092
Adjacent sequences: A056898 A056899 A056900 * A056902 A056903 A056904
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KEYWORD
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nonn
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AUTHOR
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Lekraj Beedassy, Feb 12 2002
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EXTENSIONS
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Edited and extended by Robert G. Wilson v, Feb 18 2002
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STATUS
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approved
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