A382932 a(n) is the altitude of the Pythagorean triangle (A046083(A382931(n)), A046084(A382931(n)), A009000(A382931(n))).

12, 24, 36, 48, 60, 72, 60, 84, 96, 108, 120, 132, 120, 144, 156, 120, 168, 180, 192, 204, 216, 228, 240, 180, 252, 264, 276, 240, 288, 300, 168, 312, 324, 240, 336, 348, 360, 372, 384, 396, 420, 300, 408, 360, 420, 432, 444, 456, 468, 480, 360, 492, 504, 516

Offset

1,1

Comments

All terms are divisible by 12. Proof: (Start)

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.

With a = p^2 - q^2 and b = 2*p*q follows h = 2*k*p*q*(p^2 - q^2) = k*2*p*q*(p + q)*(p - q), where p > q > 0, gcd(p,q) = 1 and p or q is even.

It is to show that p*q*(p + q)*(p - q) is divisible by 6. Since p or q is divisible by 2, it remains to show that p*q*(p + q)*(p - q) is divisible by 3.

If 3 is a divisor of p or q, p*q is divisible by 3. If p mod 3 = 1 and q mod 3 = 2 or p mod 3 = 2 and q mod 3 = 1, then p + q is divisible by 3. If p mod 3 = q mod 3 = 1 or p mod 3 = q mod 3 = 2, then p - q is divisible by 3.

It follows that all terms are divisible by 12. (End)

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Pythagorean Triple

Formula

a(n) = A046083(A382931(n))*A046084(A382931(n))/A009000(A382931(n)).

Example

a(1) = 12 because the Pythagorean triangle (A046083(A382931(1)), A046084(A382931(1)), A009000(A382931(1))) = (A046083(7), A046084(7), A009000(7)) = (15, 20, 25) has the integer altitude 15*20/25 = 12.

Maple

A382932:=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), L[k, 3]]
        fi
    od;
    return op(M)
end proc;
A382932(1075);

Crossrefs

Cf. A008594, A009000, A046083, A046084, A382931.

Sequence in context: A336657 A066085 A340511 * A094529 A270571 A044852

Adjacent sequences: A382929 A382930 A382931 * A382933 A382934 A382935

Keyword

nonn

Author

Felix Huber, Apr 13 2025

Status

approved