A009177 - OEIS

%I #41 Aug 17 2018 18:46:39

%S 25,50,65,75,85,100,125,130,145,150,169,170,175,185,195,200,205,221,

%T 225,250,255,260,265,275,289,290,300,305,325,338,340,350,365,370,375,

%U 377,390,400,410,425,435,442,445,450,455,475,481,485,493,500,505,507,510,520,525

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

%C 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_

%C 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

%C Numbers appearing more than once in A009000. - _Sean A. Irvine_, Apr 20 2018

%H Robert Israel, <a href="/A009177/b009177.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F 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]

%e 25^2 = 24^2 + 7^2 = 20^2 + 15^2.

%e 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_

%p filter:= proc(n) add(`if` (t[1] mod 4 = 1, t[2],0), t = ifactors(n)[2]) >= 2 end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Dec 21 2015

%t 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 *)

%Y Cf. A004431, A009000, A118882, A121387.

%K nonn

%O 1,1

%A _David W. Wilson_