cp's OEIS Frontend

A386308 Long legs of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2) ordered by increasing hypotenuse (A386307), where u and v are positive integers.

%I A386308 #9 Aug 23 2025 23:39:56
%S A386308 12,20,24,28,36,40,45,44,48,52,60,56,60,72,72,68,75,63,84,76,80,90,84,
%T A386308 88,105,92,105,96,120,120,104,120,108,112,132,105,116,120,144,124,144,
%U A386308 135,132,156,136,150,126,140,168,168,180,148,175,165,152,156,168,180
%N A386308 Long legs of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2) ordered by increasing hypotenuse (A386307), where u and v are positive integers.
%C A386308 In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.
%H A386308 Felix Huber, <a href="/A386308/b386308.txt">Table of n, a(n) for n = 1..10000</a>
%H A386308 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A386308 a(n) = sqrt(A386307(n)^2 - A386309(n)^2).
%F A386308 {A046084(n)} = {a(n)} union {A046087(n)} union {A386944(n)}.
%e A386308 The Pythagorean triple (9, 12, 15) does not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), because 15 is not a sum of two nonzero squares. Therefore 12 is a term.
%p A386308 A386308:=proc(N) # To get all terms with hypotenuses <= N
%p A386308     local i,l,m,u,v,r,x,y,z;
%p A386308     l:={};
%p A386308     m:={};
%p A386308     for u from 2 to floor(sqrt(N-1)) do
%p A386308         for v to min(u-1,floor(sqrt(N-u^2))) do
%p A386308             x:=min(2*u*v,u^2-v^2);
%p A386308             y:=max(2*u*v,u^2-v^2);
%p A386308             z:=u^2+v^2;
%p A386308             m:=m union {[z,y,x]};
%p A386308             if gcd(u,v)=1 and is(u-v,odd) then
%p A386308                 l:=l union {seq([i*z,i*y,i*x],i=1..N/z)}
%p A386308             fi
%p A386308         od
%p A386308     od;
%p A386308     r:=l minus m;
%p A386308     return seq(r[i,2],i=1..nops(r));
%p A386308 end proc;
%p A386308 A386308(1000);
%Y A386308 Subsequence of A046084.
%Y A386308 Cf. A366675, A380073, A386307, A386309, A386944.
%K A386308 nonn,new
%O A386308 1,1
%A A386308 _Felix Huber_, Aug 19 2025