decompwlj 3D

Digitally balanced numbers: positive numbers that in base 2 have the same number of 0's as 1's

A031443 on the OEIS · family binary rule · also known as Balanced binary numbers

Weight–level plate of Digitally balanced numbers: positive numbers that in base 2 have the same number of 0's as 1's
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 viewerA031443 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,991
Level class, k > L11,830 · 11.83 %
Weight class, k ≤ L88,161 · 88.17 %
Ties, k = L15
On the level line L = 17,742
Forced level, l ≤ d²0
Range of a(n)2 … 877,622
Range of the jump d1 … 263,167
Largest weight k, level L877,619, 438,810

As many 0 bits as 1 bits. The level share is 11.83 %; L = 1 holds 65 % 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.