Numbers m that are the hypotenuse of exactly 13 distinct integer-sided right triangles, i.e., m^2 can be written as a sum of two squares in 13 ways

A097102 on the OEIS · family quadratic form

Weight–level plate of Numbers m that are the hypotenuse of exactly 13 distinct integer-sided right triangles, i.e., m^2 can be written as a sum of two squares in 13 ways — Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.

Open in the 3-D viewer A097102 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,999
Level class, k > L	23,642 · 23.64 %
Weight class, k ≤ L	76,357 · 76.36 %
Ties, k = L	6
On the level line L = 1	4,511
Forced level, l ≤ d²	83
Range of a(n)	1,105 … 2,928,488
Range of the jump d	1 … 780
Largest weight k, level L	2,928,481, 1,461,444

231 different gaps occur, from 1 to 780; the level share is 23.64 %.

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.