decompwlj 3D

The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2

A007088 on the OEIS · family binary rule · also known as Binary expansion read in decimal

Weight–level plate of The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2
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 viewerA007088 on the OEIS
Terms65,535 (n = 0 … 65,534)
Decomposable (a > 2d)65,519
Level class, k > L8,298 · 12.67 %
Weight class, k ≤ L57,221 · 87.33 %
Ties, k = L1
On the level line L = 12,538
Forced level, l ≤ d²367
Range of a(n)0 … 1,111,111,111,111,110
Range of the jump d1 … 888,888,888,888,889
Largest weight k, level L1,111,111,111,110,109, 370,370,370,370,333

n in binary, read in decimal. The gap is (8*10^j + 1)/9 = 1, 9, 89, 889, ... where j counts the trailing 1 bits of n: sixteen values here, and the plane splits into clean diagonal lines. Capped at 65,535 terms, since a(65,536) = 10^16 passes 2^53.

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.