decompwlj 3D

Numbers that are not powers of primes p^k (k >= 0); complement of A000961

A024619 on the OEIS · family complement · also known as Not prime powers

Weight–level plate of Numbers that are not powers of primes p^k (k >= 0); complement of A000961
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 viewerA024619 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L10,591 · 10.59 %
Weight class, k ≤ L89,408 · 89.41 %
Ties, k = L65
On the level line L = 19,143
Forced level, l ≤ d²0
Range of a(n)6 … 110,614
Range of the jump d1 … 4
Largest weight k, level L110,609, 55,306

Numbers with at least two distinct prime factors. Gaps of 1 to 4; the plane is close to the naturals' (10.59 % level).

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.