decompwlj 3D

Numbers that contain only 1's, 2's and 3's

A007932 on the OEIS · family digit rule

Weight–level plate of Numbers that contain only 1's, 2's and 3's
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 viewerA007932 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,979
Level class, k > L10,692 · 10.69 %
Weight class, k ≤ L89,287 · 89.31 %
Ties, k = L1
On the level line L = 13,855
Forced level, l ≤ d²480
Range of a(n)1 … 11,231,311,131
Range of the jump d1 … 7,777,777,778
Largest weight k, level L11,231,311,111, 5,615,655,565

11 different gaps occur, from 1 to 7,777,777,778; the level share is 10.69 %; L = 5 holds 43 % of the level class; 21 terms do not decompose.

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.