Numbers with an even number of prime divisors (counted with multiplicity); numbers k such that the Liouville function lambda(k) (A008836) is positive

A028260 on the OEIS · family multiplicative · also known as Even number of prime factors

Weight–level plate of Numbers with an even number of prime divisors (counted with multiplicity); numbers k such that the Liouville function lambda(k) (A008836) is positive — 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 viewer A028260 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,997
Level class, k > L	12,780 · 12.78 %
Weight class, k ≤ L	87,217 · 87.22 %
Ties, k = L	39
On the level line L = 1	8,910
Forced level, l ≤ d²	4
Range of a(n)	1 … 200,305
Range of the jump d	1 … 17
Largest weight k, level L	200,293, 100,152

Omega(n) even (Liouville lambda = 1), the complement of the odd-Omega numbers (12.82 % level there). The level share is 12.78 %; L = 1 holds 70 % 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.