decompwlj 3D

Numbers that are the sum of 3 but no fewer nonzero squares

A000419 on the OEIS · family quadratic form

Weight–level plate of Numbers that are the sum of 3 but no fewer nonzero squares
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 viewerA000419 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L12,593 · 12.59 %
Weight class, k ≤ L87,405 · 87.41 %
Ties, k = L3
On the level line L = 18,892
Forced level, l ≤ d²1
Range of a(n)3 … 166,907
Range of the jump d1 … 5
Largest weight k, level L166,871, 83,450

5 different gaps occur, from 1 to 5; the level share is 12.59 %; L = 1 holds 71 % of the level class.

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.