decompwlj 3D

Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative

A026424 on the OEIS · family multiplicative · also known as Odd number of prime factors

Weight–level plate of Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative
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 viewerA026424 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L12,821 · 12.82 %
Weight class, k ≤ L87,177 · 87.18 %
Ties, k = L50
On the level line L = 19,087
Forced level, l ≤ d²3
Range of a(n)2 … 199,726
Range of the jump d1 … 16
Largest weight k, level L199,721, 99,861

Omega(n) odd (Liouville lambda = -1). Gaps of 1 to 16; the level share is 12.82 %.

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.