decompwlj 3D

Numbers whose binary representation ends in an odd number of zeros

A036554 on the OEIS · family binary rule · also known as Odd number of trailing 0 bits

Weight–level plate of Numbers whose binary representation ends in an odd number of zeros
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 viewerA036554 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L13,875 · 13.88 %
Weight class, k ≤ L86,124 · 86.12 %
Ties, k = L0
On the level line L = 12
Forced level, l ≤ d²4
Range of a(n)2 … 300,000
Range of the jump d2 … 4
Largest weight k, level L149,971, 99,996

An odd number of trailing 0 bits, the complement of A003159. Every term is twice a term of A003159. Both have exactly 13,875 level terms, all on L = 1 there and here almost all on L = 2. The level share is 13.88 %; L = 2 holds 100 % of the level class; there are no ties.

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.