A009177 Numbers that are the hypotenuses of more than one Pythagorean triangle.

25, 50, 65, 75, 85, 100, 125, 130, 145, 150, 169, 170, 175, 185, 195, 200, 205, 221, 225, 250, 255, 260, 265, 275, 289, 290, 300, 305, 325, 338, 340, 350, 365, 370, 375, 377, 390, 400, 410, 425, 435, 442, 445, 450, 455, 475, 481, 485, 493, 500, 505, 507, 510, 520, 525

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Also, hypotenuses of Pythagorean triangles in Pythagorean triples (a, b, c, a < b < c) such that a and b are the hypotenuses of Pythagorean triangles, where the Pythagorean triples (x1, y1, a) and (x2, y2, b) are similar triangles. Sequence gives c values. - Naohiro Nomoto

Any multiple of a term of this sequence is also a term. The primitive elements are the products of two primes, not necessarily distinct, that are == 1 (mod 4): A121387. - Franklin T. Adams-Watters, Dec 21 2015

Numbers appearing more than once in A009000. - Sean A. Irvine, Apr 20 2018

Examples

			25^2 = 24^2 + 7^2 = 20^2 + 15^2.
E.g., (a = 15, b = 20, c = 25, a^2 + b^2=c^2); 15 and 20 are the hypotenuses of Pythagorean triangles. The Pythagorean triples (9, 12, 15) and (12, 16, 20) are similar triangles. So c = 25 is in the sequence. - _Naohiro Nomoto_
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A004431, A009000, A118882, A121387.

Programs

Maple

filter:= proc(n) add(`if` (t[1] mod 4 = 1, t[2],0), t = ifactors(n)[2]) >= 2 end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 21 2015

Mathematica

f[n_] := Module[{i = 0, k = 0}, Do[If[Sqrt[n^2 - i^2] == IntegerPart[Sqrt[n^2 - i^2]], k++], {i, n - 1, 1, -1}]; k];
lst = {}; Do[If[f[n] > 2, AppendTo[lst, n]], {n, 4*5!}];
lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)

Formula

Of the form b(i)*b(j)*k, where b(n) is A004431(n). Numbers whose prime factorization includes at least 2 (not necessarily distinct) primes congruent to 1 mod 4. - Franklin T. Adams-Watters, May 03 2006. [Typo corrected by Ant King, Jul 17 2008]