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

1,1

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sums of squares

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]

Example

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

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

Clear[lst, f, n, i, k]; 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 *)

Crossrefs

Cf. A004431, A009000, A118882, A121387.

Sequence in context: A236834 A033902 A354379 * A118882 A085625 A116490

Adjacent sequences: A009174 A009175 A009176 * A009178 A009179 A009180

Keyword

nonn

Author

David W. Wilson

Status

approved