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A056901 Least semiperimeter s of primitive Pythagorean triangle with inradius n.

%I A056901 #19 Feb 16 2025 08:32:43
%S A056901 6,15,20,45,42,35,72,153,110,63,156,77,210,99,88,561,342,143,420,117,
%T A056901 130,195,600,209,702,255,812,165,930,187,1056,2145,238,399,204,221,
%U A056901 1482,483,304,273,1806,247,1980,285,266,675,2352,665,2550,783,460,357
%N A056901 Least semiperimeter s of primitive Pythagorean triangle with inradius n.
%C A056901 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).
%C A056901 Moreover, a primitive Pythagorean triangle has area n*a(n).
%D A056901 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.
%D A056901 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.
%H A056901 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiperimeter.html">Semiperimeter</a>
%H A056901 Wm. H. Richardson, <a href="http://www.math.wichita.edu/~richardson/Inradius.html">The inradius of a Right Triangle with Integral Sides</a>
%F A056901 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).
%t A056901 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} ]
%Y A056901 Cf. A014498.
%K A056901 nonn
%O A056901 1,1
%A A056901 _Lekraj Beedassy_, Feb 12 2002
%E A056901 Edited and extended by _Robert G. Wilson v_, Feb 18 2002