A396768 a(n) is the number of primes p < q = prime(n) for which there exists an integer k such that (p, q, k) is a Heronian triangle.

0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 4, 1, 0, 0, 5, 2, 0, 0, 1, 0, 1, 0, 4, 0, 0, 0, 0, 2, 0, 1, 1, 0, 0, 0, 0, 3, 2, 0, 5, 0, 1, 0, 1, 0, 3, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 3, 4, 0, 0, 3, 0, 0, 1, 3, 3, 3, 0, 3, 0, 1, 0, 0, 0, 5

Offset

1,6

Comments

Let s = (p + q + k)/2. Then (p, q, k) is Heronian iff |p - q| < k < p + q, p and q are odd primes, k is even (so that s, s - p, s - q, s - k are integers), and s*(s - p)*(s - q)*(s - k) is a perfect square.

For a fixed pair of odd primes (p, q), the number of admissible k is 0, 1, or 2. This sequence only records existence (p is counted once if at least one k exists).

If p == q == 3 (mod 4), then no k exists. If q - p == 4 (mod 8), then no k exists.

Links

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

Eric Weisstein's World of Mathematics, Heronian Triangle

Example

a(6) = 2, since for q = prime(6) = 13 two Heronian triangles with p < 13 exist: (5, 12, 13) and (11, 13, 20).

Maple

f := proc(x, y) option remember;
    local k, l, r, s, t, u, v;
    if x = 2 then return false end if;
    if irem(x, 4) = 3 and irem(y, 4) = 3 then return false end if;
    if irem(x - y, 8) = 4 then return false end if;
    l := max(2, abs(x - y) + 1); if type(l, odd) then l := l + 1 end if;
    u := x + y - 1;
    for k from l by 2 to u do
        s := (x + y + k)/2; v := (x + y - k)/2; r := (-x + y + k)/2; t := (x - y + k)/2;
        if issqr(s*v*r*t) then return true end if;
    end do;
    false
end proc:
A := proc(n) option remember;
    local a, i, p, q;
    q := ithprime(n); a := 0;
    for i from 2 to n - 1 do
        p := ithprime(i);
        if f(p, q) then a := a + 1 end if;
    end do;
    a
end proc:
seq(A(n), n = 1 .. 88);

Crossrefs

Cf. A123323, A229746, A239246, A395186, A396755, A396756.

Sequence in context: A336107 A350251 A367783 * A363808 A227761 A037188

Adjacent sequences: A396765 A396766 A396767 * A396769 A396770 A396771

Keyword

nonn

Author

Felix Huber, Jun 05 2026

Status

approved