decompwlj 3D

Numbers having equal length of longest contiguous block of zeros and ones in binary expansion

A090050 on the OEIS · family binary rule

Weight–level plate of Numbers having equal length of longest contiguous block of zeros and ones in binary expansion
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 viewerA090050 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L14,034 · 14.03 %
Weight class, k ≤ L85,962 · 85.97 %
Ties, k = L33
On the level line L = 18,310
Forced level, l ≤ d²36
Range of a(n)2 … 550,031
Range of the jump d1 … 2,045
Largest weight k, level L550,009, 275,004

65 different gaps occur, from 1 to 2,045; the level share is 14.03 %; L = 1 holds 59 % 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.