A396756 a(n) is the number of Heronian triangles (p, q, k) with primes p <= q < k, where k = A396755(n).

1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset

1,5

Links

Table of n, a(n) for n=1..88.

Eric Weisstein's World of Mathematics, Heronian Triangle

Example

The a(15) = 6 Heronian triangles are (61, 61, 120), (31, 97, 120), (29, 101, 120), (13, 109, 120), (109, 109, 120) and (17, 113, 120), where A396755(15) = 120.

Maple

g := proc(k) option remember;
    local a, m, p, q, u, v;
    if irem(k, 2) = 1 or k < 6 then return 0 end if;
    a := 0;
    q := nextprime(iquo(k + 1, 2) - 1);
    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
                a := a + 1
            end if;
            p := nextprime(p)
        end do;
        q := nextprime(q)
    end do;
    a
end proc:
A396756 := proc(n) option remember;
    local a, k, t;
    t := 0; k := 4;
    do
        k := k + 2;
        a := g(k);
        if a > 0 then t := t + 1 end if;
        if t = n then return a end if;
    end do;
end proc:
seq(A396756(n), n = 1 .. 88);

Crossrefs

Cf. A054875, A123323, A239246, A395186, A396755.

Sequence in context: A286132 A105551 A305195 * A073772 A164562 A058188

Adjacent sequences: A396751 A396752 A396755 * A396757 A396758 A396759

Keyword

nonn

Author

Felix Huber, Jun 05 2026

Status

approved