decompwlj 3D

Numbers that are not squares (or, the nonsquares)

A000037 on the OEIS · family complement · also known as Non-squares

Weight–level plate of Numbers that are not squares (or, the nonsquares)
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 viewerA000037 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L9,645 · 9.65 %
Weight class, k ≤ L90,353 · 90.35 %
Ties, k = L65
On the level line L = 19,593
Forced level, l ≤ d²0
Range of a(n)2 … 100,316
Range of the jump d1 … 2
Largest weight k, level L100,313, 50,157

Non-squares: gaps of 1, and 2 across each square. The plane is the naturals' almost unchanged (9.65 % level against 9.59 %); the gap-2 terms add a short line at L = 2.

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.