decompwlj 3D

Numbers that are the sum of 4 but no fewer nonzero squares

A004215 on the OEIS · family quadratic form · also known as Not sums of three squares

Weight–level plate of Numbers that are the sum of 4 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 viewerA004215 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L23,707 · 23.71 %
Weight class, k ≤ L76,291 · 76.29 %
Ties, k = L46
On the level line L = 112,235
Forced level, l ≤ d²6
Range of a(n)7 … 600,015
Range of the jump d1 … 8
Largest weight k, level L599,999, 299,999

4^a(8b + 7), the complement of the sums of three squares. Half the gaps are 8. The level share is 23.71 %, with L = 1 holding half 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.