decompwlj 3D

Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true

A052485 on the OEIS · family multiplicative · also known as Weak numbers (not powerful)

Weight–level plate of Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true
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 viewerA052485 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L9,621 · 9.62 %
Weight class, k ≤ L90,377 · 90.38 %
Ties, k = L66
On the level line L = 19,565
Forced level, l ≤ d²1
Range of a(n)2 … 100,622
Range of the jump d1 … 3
Largest weight k, level L100,621, 50,310

Numbers that are not powerful (some prime divides them exactly once). Gap 1 on 99.4 % of the terms. The plane is the naturals' almost unchanged: 9.62 % level against 9.59 %.

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.