decompwlj 3D

Odious numbers: numbers with an odd number of 1's in their binary expansion

A000069 on the OEIS · family binary rule · also known as Odious numbers

Weight–level plate of Odious numbers: numbers with an odd number of 1's in their 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 viewerA000069 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L11,011 · 11.01 %
Weight class, k ≤ L88,986 · 88.99 %
Ties, k = L86
On the level line L = 19,285
Forced level, l ≤ d²1
Range of a(n)1 … 199,999
Range of the jump d1 … 3
Largest weight k, level L199,961, 99,996

Numbers with an odd number of 1 bits. Gaps are 1, 2 and 3, each on a third of the terms. The level share is 11.01 %, next to the evil numbers' 11.39 %, with 86 ties each. 84 % of the level class sits on L = 1 and the rest on L = 3.

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.