decompwlj 3D

Numbers of the form k + wt(k) for exactly two distinct k, where wt(k) = A000120(k) is the binary weight of k

A230091 on the OEIS · family binary rule

Weight–level plate of Numbers of the form k + wt(k) for exactly two distinct k, where wt(k) = A000120(k) is the binary weight of k
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 viewerA230091 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L17,634 · 17.63 %
Weight class, k ≤ L82,365 · 82.37 %
Ties, k = L28
On the level line L = 18,305
Forced level, l ≤ d²7
Range of a(n)5 … 433,831
Range of the jump d1 … 11
Largest weight k, level L433,813, 216,644

10 different gaps occur, from 1 to 11; the level share is 17.63 %; L = 1 holds 47 % 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.