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A386309 Short 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 A386309 #9 Aug 23 2025 23:40:26
%S A386309 9,15,18,21,15,30,24,33,36,39,25,42,45,21,30,51,40,60,35,57,60,48,63,
%T A386309 66,36,69,56,72,27,35,78,50,81,84,55,100,87,90,42,93,60,84,99,65,102,
%U A386309 80,120,105,49,70,33,111,60,88,114,117,99,75,48,120,140,96,123,45
%N A386309 Short 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 A386309 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 A386309 Felix Huber, <a href="/A386309/b386309.txt">Table of n, a(n) for n = 1..10000</a>
%H A386309 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A386309 a(n) = sqrt(A386307(n)^2 - A386308(n)^2).
%F A386309 {A046083(n)} = {a(n)} union {A046086(n)} union {A386945(n)}.
%e A386309 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 A386309 A386309:=proc(N) # To get all terms with hypotenuses <= N
%p A386309     local i,l,m,u,v,r,x,y,z;
%p A386309     l:={};
%p A386309     m:={};
%p A386309     for u from 2 to floor(sqrt(N-1)) do
%p A386309         for v to min(u-1,floor(sqrt(N-u^2))) do
%p A386309             x:=min(2*u*v,u^2-v^2);
%p A386309             y:=max(2*u*v,u^2-v^2);
%p A386309             z:=u^2+v^2;
%p A386309             m:=m union {[z,y,x]};
%p A386309             if gcd(u,v)=1 and is(u-v,odd) then
%p A386309                 l:=l union {seq([i*z,i*y,i*x],i=1..N/z)}
%p A386309             fi
%p A386309         od
%p A386309     od;
%p A386309     r:=l minus m;
%p A386309     return seq(r[i,3],i=1..nops(r));
%p A386309 end proc;
%p A386309 A386309(1000);
%Y A386309 Subsequence of A046083.
%Y A386309 Cf. A366674, A380074, A386307, A386308, A386945.
%K A386309 nonn,new
%O A386309 1,1
%A A386309 _Felix Huber_, Aug 19 2025