OFFSET
1,1
COMMENTS
Primitive cases only: gcd(x, y, z, m) = 1. All others arise by scaling a primitive triangle by an integer k > 1.
The middle and shortest sides are in A394367 and A394368, respectively. The corresponding median values m are in A394369.
A triangle (x <= y <= z) is included if there exists a median with integer length m = sqrt(2*y^2 + 2*z^2 - x^2)/2 (or cyclically at y or z), and every integral median m satisfies gcd(x, y, z, m) = 1.
From the parity condition in 4*m^2 = 2*y^2 + 2*z^2 - x^2 it follows that any side whose median is integral must be even. Thus the median divides the triangle into two integer-sided triangles. If medians to two distinct sides were integral, then x, y, z would all be even; under the gcd condition this leads to infinite 2-adic descent unless the two medians coincide. Hence at most one distinct integral median length can occur (in the isosceles case two medians may exist geometrically, but they are equal).
LINKS
Felix Huber, Table of n, a(n) for n = 1..10098
Felix Huber, Maple program
Eric Weisstein's World of Mathematics, Triangle Median
EXAMPLE
6 is a term because in the triangle (x, y, z) = (5, 5, 6) the median to z = 6 has length m = 4. It splits z into 3 and 3, yielding two congruent triangles (3, 4, 5), and gcd(5, 5, 6, 4) = 1.
9 is a term because in the triangle (x, y, z) = (7, 8, 9) the median to y = 8 has length m = 7. It splits y into 4 and 4, yielding triangles (4, 7, 7) and (4, 7, 9), and gcd(7, 8, 9, 7) = 1.
14 is a term because in the triangle (x, y, z) = (8, 14, 14) the medians to y = 14 and z = 14 both have length m = 9. Each splits the side into 7 and 7, yielding triangles (7, 8, 9) and (7, 9, 14)., and gcd(8, 14, 14, 9) = 1.
MAPLE
# See Huber link.
CROSSREFS
KEYWORD
nonn
AUTHOR
Felix Huber, Mar 20 2026
STATUS
approved