decompwlj 3D

Numbers that are not perfect powers

A007916 on the OEIS · family powers · also known as Non-perfect powers

Weight–level plate of Numbers that are not perfect powers
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 viewerA007916 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L9,651 · 9.65 %
Weight class, k ≤ L90,347 · 90.35 %
Ties, k = L64
On the level line L = 19,598
Forced level, l ≤ d²1
Range of a(n)2 … 100,367
Range of the jump d1 … 3
Largest weight k, level L100,363, 50,183

Not a perfect power. Gaps 1 to 3: the naturals' plane. The level share is 9.65 %; L = 1 holds 99 % 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.