decompwlj 3D

Numbers not divisible by 3

A001651 on the OEIS · family forced divisor

Weight–level plate of Numbers not divisible by 3
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 viewerA001651 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L3 · 0.00 %
Weight class, k ≤ L99,995 · 100.00 %
Ties, k = L2
On the level line L = 12
Forced level, l ≤ d²1
Range of a(n)1 … 149,999
Range of the jump d1 … 2
Largest weight k, level L3, 74,997

The gaps alternate 1, 2 and 3 divides l at every term. Since d <= 2, the forced divisor 3 is always eligible, so k is 2 or 3 at every decomposable term (24,999 and 74,999 of them) and the cloud collapses to two vertical lines. Only a = 4, 5 and 8 are level-classified: the most weight-dominated sequence in the atlas.

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.