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A379600 a(n) is the semiperimeter of the primitive Pythagorean triangle (x(n), y(n), z(n)) with x(n) < y(n) < z(n) and x(n) > x(n-1), y(n) > y(n-1), z(n) > z(n-1), which has the smallest perimeter (if there are several triangles with smallest perimeter: the one of these with the smallest area), starting from a(1) = (3 + 4 + 5)/2 = 6.

%I A379600 #34 Mar 05 2025 18:59:59
%S A379600 6,15,20,35,63,77,99,104,130,165,204,247,266,336,345,391,425,450,513,
%T A379600 580,609,651,713,805,825,888,945,1036,1107,1204,1271,1376,1457,1530,
%U A379600 1617,1645,1764,1887,1961,2014,2090,2280,2337,2419,2537,2562,2684,2772,2990,3149
%N A379600 a(n) is the semiperimeter of the primitive Pythagorean triangle (x(n), y(n), z(n)) with x(n) < y(n) < z(n) and x(n) > x(n-1), y(n) > y(n-1), z(n) > z(n-1), which has the smallest perimeter (if there are several triangles with smallest perimeter: the one of these with the smallest area), starting from a(1) = (3 + 4 + 5)/2 = 6.
%C A379600 Conjecture: There is no primitive Pythagorean triangle that has a smaller semiperimeter than a(n) that can be drawn around the primitive Pythagorean triangle (x(n-1), y(n-1), z(n-1)) without touching it.
%C A379600 Subsequence of A020886.
%H A379600 Felix Huber, <a href="/A379600/b379600.txt">Table of n, a(n) for n = 1..3257</a>
%H A379600 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimitivePythagoreanTriple.html">Primitive Pythagorean Triple</a>
%H A379600 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%e A379600 (8, 15, 17) is the primitive Pythagorean triangle with semiperimeter a(3) = 20. (20, 21, 29) is the primitive Pythagorean triangle with semiperimeter a(4) = 35 because 20 > 8, 21 > 15, 29 > 17 and there is no other primitive Pythagorean triangle with perimeter <= 70 satisfying this criterium. For example, the primitive Pythagorean triangle (7, 24, 25) has the perimeter 56 but 7 < 8.
%p A379600 A379600:=proc(S) # to get all terms <= S
%p A379600     local p,q,i,L,M;
%p A379600     L:=[];
%p A379600     M:=[[3,4,5,6,6]];
%p A379600     for p from 3 to floor((sqrt(4*S+1)-1)/2) do
%p A379600         for q to min(p-1,S/p-p) do
%p A379600             if gcd(p,q)=1 and is(p-q,odd) then
%p A379600                 L:=[op(L),[min(p^2-q^2,2*p*q),max(p^2-q^2,2*p*q),p^2+q^2,p*(p+q),(p^2-q^2)*p*q]];
%p A379600             fi
%p A379600         od
%p A379600     od;
%p A379600     L:=sort(sort(L,(x,y)->x[5]<=y[5]),(x,y)->x[4]<=y[4]);
%p A379600     for i in L do
%p A379600         if i[1]>M[nops(M),1] and i[2]>M[nops(M),2] and i[3]>M[nops(M),3] then
%p A379600             M:=[op(M),i]
%p A379600         fi
%p A379600     od;
%p A379600     return seq(M[i,4],i=1..nops(M))
%p A379600 end proc;
%p A379600 A379600(3149);
%p A379600 # 3 lines above: change 4 to 3 for hypotenuses, to 2 for long legs and to 1 for short legs, to 5 for areas
%Y A379600 Cf. A010814, A020886, A118858.
%K A379600 nonn
%O A379600 1,1
%A A379600 _Felix Huber_, Feb 15 2025