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A386943 Ordered hypotenuses of nonprimitive Pythagorean triples of the form (u^2 - v^2, 2*u*v, u^2 + v^2), where u and v are positive integers.

%I A386943 #8 Sep 02 2025 22:53:30
%S A386943 10,20,26,34,40,45,50,52,58,68,74,80,82,90,100,104,106,116,117,122,
%T A386943 125,130,130,136,146,148,153,160,164,170,170,178,180,194,200,202,208,
%U A386943 212,218,225,226,232,234,244,245,250,250,260,260,261,272,274,290,290,292,296
%N A386943 Ordered hypotenuses of nonprimitive Pythagorean triples of the form (u^2 - v^2, 2*u*v, u^2 + v^2), where u and v are positive integers.
%C A386943 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 A386943 A101930(n) gives the total number of Pythagorean triples <= 10^n.
%C A386943                number of terms <= h     total number of
%C A386943        h       in this sequence         hypotenuses <= h      percentage
%C A386943       10                 1                    2                 50.0 %
%C A386943      100                15                   52                 28.8 %
%C A386943     1000               209                  881                 23.7 %
%C A386943    10000              2249                12471                 18.0 %
%C A386943   100000             23086               161436                 14.3 %
%H A386943 Felix Huber, <a href="/A386943/b386943.txt">Table of n, a(n) for n = 1..10000</a>
%H A386943 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A386943 a(n) = sqrt(A386944(n)^2 + A386945(n)^2).
%F A386943 {A009000(n)} = {a(n)} union {A020882(n)} union {A386307(n)}.
%e A386943 The nonprimitive Pythagorean triple (6, 8, 10) is of the form (u^2 - v^2, 2*u*v, u^2 + v^2): From u = 3 and v = 1 follows u^2 - v^2 = 8 (long leg), 2*u*v = 6 (short leg), u^2 - v^2 = 10 (hypotenuse). Therefore, 10 is a term.
%p A386943 A386943:=proc(N) # To get all hypotenuses <= N
%p A386943     local i,l,u,v;
%p A386943     l:=[];
%p A386943     for u from 2 to floor(sqrt(N-1)) do
%p A386943         for v to min(u-1,floor(sqrt(N-u^2))) do
%p A386943             if gcd(u,v)>1 or is(u-v,even) then
%p A386943                 l:=[op(l),[u^2+v^2,max(2*u*v,u^2-v^2),min(2*u*v,u^2-v^2)]]
%p A386943             fi
%p A386943         od
%p A386943     od;
%p A386943     l:=sort(l);
%p A386943     return seq(l[i,1],i=1..nops(l));
%p A386943 end proc;
%p A386943 A386943(296);
%Y A386943 Subsequence of A009000.
%Y A386943 Cf. A020882, A101930, A366428, A380072, A386307, A386944, A386945.
%K A386943 nonn,easy,new
%O A386943 1,1
%A A386943 _Felix Huber_, Aug 24 2025