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A380072 Ordered hypotenuses of Pythagorean triangles having legs that add up to a square.

%I A380072 #10 Jan 25 2025 23:01:34
%S A380072 35,41,140,164,205,221,315,369,389,391,560,656,689,775,820,875,884,
%T A380072 1025,1189,1260,1476,1556,1564,1565,1625,1715,1739,1781,1845,1855,
%U A380072 1989,2009,2240,2624,2756,2835,3100,3280,3321,3500,3501,3519,3536,3865,3869,4100,4105
%N A380072 Ordered hypotenuses of Pythagorean triangles having legs that add up to a square.
%C A380072 Corresponding long legs in A380073, short legs in A380074.
%C A380072 Subsequence of A009000 and supersequence of A088319.
%H A380072 Felix Huber, <a href="/A380072/b380072.txt">Table of n, a(n) for n = 1..10001</a>
%H A380072 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%e A380072 35 is in the sequence because 21^2 + 28^2 = 35^2 and 21 + 28 = 7^2.
%e A380072 206125 is twice in the sequence because 31525^2 + 203700^2 = 206125^2 and 31525 + 203700 = 485^2 as well as 94588^2 + 183141^2 = 206125^2 and 94588 + 183141 = 527^2.
%p A380072 # Calculates the first 10001 terms
%p A380072 A380072:=proc(M)
%p A380072     local i,m,p,q,r,w,L;
%p A380072     L:=[];
%p A380072     m:=M^2+2*M+2;
%p A380072     for p from 2 to M do
%p A380072         for q to p-1 do
%p A380072             if gcd(p,q)=1 and (is(p,even) or is(q,even)) then
%p A380072                 r:=1;
%p A380072                 for i in ifactors(p^2-q^2+2*p*q)[2] do
%p A380072                     if is(i[2],odd) then
%p A380072                         r:=r*i[1]
%p A380072                     fi
%p A380072                 od;
%p A380072                 w:=r*(p^2+q^2);
%p A380072                 if w<=m then
%p A380072                     L:=[op(L),seq(i^2*w,i=1..floor(sqrt(m/w)))]
%p A380072                 fi
%p A380072             fi
%p A380072         od
%p A380072     od;
%p A380072     return op(sort(L))
%p A380072 end proc;
%p A380072 A380072(4330);
%Y A380072 Cf. A000290, A007913, A009000, A088319, A379830, A380073, A380074.
%K A380072 nonn
%O A380072 1,1
%A A380072 _Felix Huber_, Jan 18 2025