decompwlj 3D

Numbers whose base-2 expansion has no run of digits with length < 2

A033015 on the OEIS · family binary rule

Weight–level plate of Numbers whose base-2 expansion has no run of digits with length < 2
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 viewerA033015 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,977
Level class, k > L19,099 · 19.10 %
Weight class, k ≤ L80,878 · 80.90 %
Ties, k = L8
On the level line L = 16,124
Forced level, l ≤ d²402
Range of a(n)3 … 30,161,151
Range of the jump d3 … 8,388,609
Largest weight k, level L30,159,839, 6,032,229

24 different gaps occur, from 3 to 8,388,609; the level share is 19.10 %; L = 1 holds 32 % of the level class; 23 terms do not decompose.

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.