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

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

Offset

1,1

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).

References

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.

Links

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

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).

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} ]

Crossrefs

Cf. A014498.

Sequence in context: A196394 A162693 A343893 * A208542 A012412 A009092

Adjacent sequences: A056898 A056899 A056900 * A056902 A056903 A056904

Keyword

nonn

Author

Lekraj Beedassy, Feb 12 2002

Extensions

Edited and extended by Robert G. Wilson v, Feb 18 2002

Status

approved