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.

6, 15, 20, 35, 63, 77, 99, 104, 130, 165, 204, 247, 266, 336, 345, 391, 425, 450, 513, 580, 609, 651, 713, 805, 825, 888, 945, 1036, 1107, 1204, 1271, 1376, 1457, 1530, 1617, 1645, 1764, 1887, 1961, 2014, 2090, 2280, 2337, 2419, 2537, 2562, 2684, 2772, 2990, 3149

Offset: 1

Author

Felix Huber, Feb 15 2025

Keywords

• nonn

Comments

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.

Subsequence of A020886.

Examples

			(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.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..3257
• Eric Weisstein's World of Mathematics, Primitive Pythagorean Triple
• Eric Weisstein's World of Mathematics, Pythagorean Triple

Crossrefs

Cf. A010814, A020886, A118858.

Programs

Maple

A379600:=proc(S) # to get all terms <= S
    local p,q,i,L,M;
    L:=[];
    M:=[[3,4,5,6,6]];
    for p from 3 to floor((sqrt(4*S+1)-1)/2) do
        for q to min(p-1,S/p-p) do
            if gcd(p,q)=1 and is(p-q,odd) then
                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]];
            fi
        od
    od;
    L:=sort(sort(L,(x,y)->x[5]<=y[5]),(x,y)->x[4]<=y[4]);
    for i in L do
        if i[1]>M[nops(M),1] and i[2]>M[nops(M),2] and i[3]>M[nops(M),3] then
            M:=[op(M),i]
        fi
    od;
    return seq(M[i,4],i=1..nops(M))
end proc;
A379600(3149);
# 3 lines above: change 4 to 3 for hypotenuses, to 2 for long legs and to 1 for short legs, to 5 for areas