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A386307 Ordered hypotenuses of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), where u and v are positive integers.

%I A386307 #13 Aug 23 2025 23:40:16
%S A386307 15,25,30,35,39,50,51,55,60,65,65,70,75,75,78,85,85,87,91,95,100,102,
%T A386307 105,110,111,115,119,120,123,125,130,130,135,140,143,145,145,150,150,
%U A386307 155,156,159,165,169,170,170,174,175,175,182,183,185,185,187,190,195,195
%N A386307 Ordered hypotenuses of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), where u and v are positive integers.
%C A386307 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.
%C A386307 A101930(n) gives the total number of Pythagorean triples <= 10^n. The percentage of triangles in this sequence increases continuously:
%C A386307                number of terms <= h     total number of
%C A386307        h       in this sequence         hypotenuses <= h      percentage
%C A386307       10                 0                    2                  0.0 %
%C A386307      100                21                   52                 40.4 %
%C A386307     1000               514                  881                 58.3 %
%C A386307    10000              8629                12471                 69.2 %
%C A386307   100000            122431               161436                 75.8 %
%H A386307 Felix Huber, <a href="/A386307/b386307.txt">Table of n, a(n) for n = 1..10000</a>
%H A386307 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A386307 a(n) = sqrt(A386308(n)^2 + A386309(n)^2).
%F A386307 {A009000(n)} = {a(n)} union {A020882(n)} union {A386943(n)}.
%e A386307 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 15 is a term.
%p A386307 A386307:=proc(N) # To get all hypotenuses <= N
%p A386307     local i,l,m,u,v,r,x,y,z;
%p A386307     l:={};
%p A386307     m:={};
%p A386307     for u from 2 to floor(sqrt(N-1)) do
%p A386307         for v to min(u-1,floor(sqrt(N-u^2))) do
%p A386307             x:=min(2*u*v,u^2-v^2);
%p A386307             y:=max(2*u*v,u^2-v^2);
%p A386307             z:=u^2+v^2;
%p A386307             m:=m union {[z,y,x]};
%p A386307             if gcd(u,v)=1 and is(u-v,odd) then
%p A386307                 l:=l union {seq([i*z,i*y,i*x],i=1..N/z)}
%p A386307             fi
%p A386307         od
%p A386307     od;
%p A386307     r:=l minus m;
%p A386307     return seq(r[i,1],i=1..nops(r));
%p A386307 end proc;
%p A386307 A386307(1000);
%Y A386307 Subsequence of A009000.
%Y A386307 Cf. A101930, A366428, A380072, A386308, A386309, A386943.
%K A386307 nonn,new
%O A386307 1,1
%A A386307 _Felix Huber_, Aug 13 2025