A396755 Numbers k such that there exist primes p <= q < k with (p, q, k) a Heronian triangle.

6, 8, 20, 24, 30, 40, 42, 56, 68, 70, 80, 90, 96, 110, 120, 130, 144, 156, 160, 170, 176, 182, 198, 210, 212, 224, 232, 234, 240, 260, 264, 280, 286, 296, 306, 312, 320, 330, 336, 350, 360, 390, 400, 408, 410, 416, 418, 420, 440, 442, 444, 450, 452, 456, 458, 462

Offset

1,1

Comments

Since no Heronian triangle has a side 2, p and q are odd primes. A triangle with three odd sides cannot be Heronian (semiperimeter half-integer), so k is even.

Links

Table of n, a(n) for n=1..56.

Eric Weisstein's World of Mathematics, Heronian Triangle

Example

6 is a term because (5, 5, 6) is a Heronian triangle.
20 is a term because (11, 13, 20) is a Heronian triangle.

Maple

f := proc(k)
    local m, p, q, u, v;
    if irem(k, 2) = 1 or k < 6 then return false end if;
    q := nextprime(iquo(k, 2));
    while q < k do
        m := max(3, k - q + 1);
        if irem(m, 2) = 0 then m := m + 1 end if;
        u := k + q;
        v := k - q;
        p := nextprime(m - 1);
        while p <= q do
            if issqr(iquo(p + u, 2)*iquo(u - p, 2)*iquo(p + v, 2)*iquo(p - v, 2)) then
                return true
            end if;
            p := nextprime(p)
        end do;
        q := nextprime(q)
    end do;
    false
end proc:
A396755 := proc(n) option remember;
    local k;
    if n = 1 then return 6 end if;
    for k from A396755(n - 1) + 2 by 2 do
        if f(k) then return k end if
    end do
end proc:
seq(A396755(n), n = 1 .. 72);

Crossrefs

Cf. A054875, A123323, A239246, A395186, A396756.

Sequence in context: A357946 A309653 A113806 * A105775 A159469 A382999

Adjacent sequences: A396750 A396751 A396752 * A396756 A396757 A396758

Keyword

nonn

Author

Felix Huber, Jun 05 2026

Status

approved