decompwlj 3D

Numbers that are the product of an even number of distinct primes

A030229 on the OEIS · family multiplicative · also known as Moebius mu(n) = 1

Weight–level plate of Numbers that are the product of an even number of distinct primes
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 viewerA030229 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L14,665 · 14.67 %
Weight class, k ≤ L85,333 · 85.33 %
Ties, k = L45
On the level line L = 110,476
Forced level, l ≤ d²5
Range of a(n)1 … 329,098
Range of the jump d1 … 30
Largest weight k, level L329,081, 164,530

Squarefree with an even number of prime factors (Moebius mu = 1). The plane is nearly the mu = -1 plane (14.67 % level against 14.60 %). The level share is 14.67 %; L = 1 holds 71 % 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.