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

%I A009177 #43 Feb 14 2025 08:04:49
%S A009177 25,50,65,75,85,100,125,130,145,150,169,170,175,185,195,200,205,221,
%T A009177 225,250,255,260,265,275,289,290,300,305,325,338,340,350,365,370,375,
%U A009177 377,390,400,410,425,435,442,445,450,455,475,481,485,493,500,505,507,510,520,525
%N A009177 Numbers that are the hypotenuses of more than one Pythagorean triangle.
%C A009177 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 A009177 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 A009177 Numbers appearing more than once in A009000. - _Sean A. Irvine_, Apr 20 2018
%H A009177 Robert Israel, <a href="/A009177/b009177.txt">Table of n, a(n) for n = 1..10000</a>
%H A009177 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A009177 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 A009177 25^2 = 24^2 + 7^2 = 20^2 + 15^2.
%e A009177 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 A009177 filter:= proc(n) add(`if` (t[1] mod 4 = 1, t[2],0), t = ifactors(n)[2]) >= 2 end proc:
%p A009177 select(filter, [$1..1000]); # _Robert Israel_, Dec 21 2015
%t A009177 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];
%t A009177 lst = {}; Do[If[f[n] > 2, AppendTo[lst, n]], {n, 4*5!}];
%t A009177 lst (* _Vladimir Joseph Stephan Orlovsky_, Aug 12 2009 *)
%Y A009177 Cf. A004431, A009000, A118882, A121387.
%K A009177 nonn
%O A009177 1,1
%A A009177 _David W. Wilson_