A394102 Shortest sides x for integer-sided triangles (x <= y <= z) with an integer internal angle bisector l splitting the triangle into two integer-sided triangles; include only tuples (x, y, z; l) with gcd(x, y, z, l) = 1; ordered by z, then y, then x.

5, 5, 6, 10, 16, 12, 5, 9, 11, 13, 14, 14, 17, 15, 17, 22, 16, 18, 24, 29, 18, 29, 13, 14, 20, 25, 28, 15, 24, 19, 14, 20, 32, 50, 35, 24, 53, 21, 19, 35, 13, 27, 22, 22, 31, 32, 33, 65, 27, 28, 37, 57, 43, 24, 17, 63, 39, 42, 65, 41, 45, 18, 41, 51, 26, 72, 48

Offset

1,1

Comments

The longest and middle sides are listed as A394100 and A394101, respectively. The corresponding l-values are in A393193.

Links

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

Example

5 is a term because in the triangle (x, y, z) = (5, 5, 6) the internal bisector l_z = 4 splits z = 6 into 3 and 3, yielding two congruent triangles (3, 4, 5), and gcd(5, 5, 6, 4) = 1.
12 is a term because in the triangle (x, y, z) = (12, 15, 18) the internal bisector l_y = 10 splits y = 15 into 6 and 9, yielding triangles (6, 10, 12) and (9, 10, 18), and gcd(12, 15, 18, 10) = 1.

Maple

# See Huber link in A394100.

Crossrefs

Cf. A181925, A182171, A393193, A394100, A394101.

Sequence in context: A191695 A298915 A008945 * A008705 A348293 A327242

Adjacent sequences: A394099 A394100 A394101 * A394103 A394104 A394105

Keyword

nonn

Author

Felix Huber, Mar 17 2026

Status

approved