decompwlj 3D

a(n) = floor(n^(3/2))

A000093 on the OEIS · family polynomial · also known as floor(n^(3/2))

Weight–level plate of a(n) = floor(n^(3/2))
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 viewerA000093 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,996
Level class, k > L46,641 · 46.64 %
Weight class, k ≤ L53,355 · 53.36 %
Ties, k = L7
On the level line L = 16,096
Forced level, l ≤ d²5
Range of a(n)0 … 31,622,302
Range of the jump d1 … 475
Largest weight k, level L31,617,559, 66,281

floor(n^(3/2)). The gap grows like sqrt(n) while l grows like n^(3/2), so l/d^2 grows like sqrt(n) and almost no term is forced level. The level share is 46.64 %.

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.