decompwlj 3D

Numbers in base 3

A007089 on the OEIS · family digit rule · also known as Ternary expansion read in decimal

Weight–level plate of Numbers in base 3
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 viewerA007089 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,979
Level class, k > L10,700 · 10.70 %
Weight class, k ≤ L89,279 · 89.30 %
Ties, k = L3
On the level line L = 13,850
Forced level, l ≤ d²360
Range of a(n)0 … 12,002,011,200
Range of the jump d1 … 7,777,777,778
Largest weight k, level L12,002,011,199, 6,001,005,560

n in base 3, read in decimal. The gaps are 1, 8, 78, 778, ..., one per count of trailing 2 digits, so the plane splits into clean lines. 21 terms fail to decompose, and the level share is 10.70 %.

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.