A382931 Numbers k for which the Pythagorean triangle (A046083(k), A046084(k), A009000(k)) has an integer altitude.

7, 19, 36, 51, 69, 88, 99, 106, 126, 147, 163, 187, 196, 208, 227, 240, 250, 273, 293, 314, 342, 361, 384, 392, 409, 434, 455, 459, 483, 504, 507, 525, 549, 552, 579, 599, 627, 649, 679, 702, 711, 718, 724, 744, 752, 775, 802, 829, 854, 879, 894, 908, 935, 960

Offset: 1

Author

Felix Huber, Apr 11 2025

Keywords

• nonn

Comments

Let (a, b, c) be a primitive Pythagorean triple. Since gcd(a, b, c) = 1, all and only the Pythagorean triples (k*c*a, k*c*b, k*c^2) have an integer altitude h = (k*c*a*k*c*b)/(k*c^2) = k*a*b, where k is a positive integer.

Examples

			7 is in the sequence because the pythagorean triangle (A046083(7), A046084(7), A009000(7)) = (15, 20, 25) has the integer altitude 15*20/25 = 12.
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Pythagorean Triple

Crossrefs

Cf. A009000, A046083, A046084, A382932.

Programs

Maple

A382931:=proc(H) # All hypotenuses <= H.
    local a,b,c,k,p,q,L,M;
    L:=[];
    M:=[];
    for p from 2 to floor(sqrt(H-1)) do
        for q to min(p-1,floor(sqrt(H-p^2))) do
            if gcd(p,q)=1 and is(p-q,odd) then
                a:=p^2-q^2;
                b:=2*p*q;
                c:=p^2+q^2;
                for k to iquo(H,c) do
                    L:=[op(L),[k*c,k*max(a,b),k*a*b/c]]
                od
            fi
        od
    od;
    L:=sort(L);
    for k to nops(L) do
        if is(L[k,3],integer) then
           M:=[op(M),k]
        fi
    od;
    return op(M)
end proc;
A382931(1075);