A394376 Longest sides z of primitive integer-sided triangles (x <= y <= z) with integer distance between circumcenter O and centroid G; ordered by increasing z, then y, then x.

1, 8, 13, 19, 35, 40, 43, 55, 65, 77, 80, 91, 91, 91, 103, 112, 119, 133, 133, 143, 151, 157, 160, 176, 187, 193, 209, 217, 221, 224, 229, 241, 247, 253, 259, 275, 277, 280, 280, 299, 301, 319, 323, 325, 341, 343, 352, 365, 377, 379, 391, 397, 403, 409, 416, 425

Offset

1,2

Comments

The middle and shortest sides are listed as A394377 and A394378, respectively. The corresponding distances OG are in A394379.

On the Euler line, the centroid G divides the segment OH (where H is the orthocenter) in the ratio 2 : 1, with OG : GH : OH = 1 : 2 : 3. Consequently, if OG is an integer, then GH = 2*OG and OH = 3*OG are also integers.

For integer OG, gcd(x, y, z) = 1 is equivalent to gcd(x, y, z, OG) = 1 since OG scales linearly with the side lengths. Therefore all cases arise by scaling the primitive cases in this sequence.

The distance between circumcenter O and centroid G is OG = sqrt(R^2 - (x^2 + y^2 + z^2)/9), where R is the circumradius. With Heron's formula H = s*(s - x)*(s - y)*(s - z) and s = (x + y + z)/2, we have R^2 = x^2*y^2*z^2/(16*H).

Links

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

Felix Huber, Maple program.

Wikipedia, Circumcircle.

Wikipedia, Euler line.

Example

1 is a term because for the triangle (x, y, z) = (1, 1, 1) we have OG = 0.
8 is a term because for the triangle (x, y, z) = (5, 7, 8) we have OG = 1.
13 is a term because for the triangle (x, y, z) = (7, 8, 13) we have OG = 5.

Maple

# See Huber link.

Crossrefs

Cf. A394377, A394378, A394379.

Sequence in context: A214723 A022954 A364411 * A359853 A070130 A070131

Adjacent sequences: A394373 A394374 A394375 * A394377 A394378 A394379

Keyword

nonn

Author

Felix Huber, Mar 24 2026

Status

approved