decompwlj 3D

a(n+1) = a(n) + sum of its digits, with a(1) = 9

A016096 on the OEIS · family digit rule

Weight–level plate of a(n+1) = a(n) + sum of its digits, with a(1) = 9
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 viewerA016096 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L17,407 · 17.41 %
Weight class, k ≤ L82,591 · 82.59 %
Ties, k = L7
On the level line L = 12
Forced level, l ≤ d²12
Range of a(n)9 … 2,614,446
Range of the jump d9 … 54
Largest weight k, level L290,471, 209,166

6 different gaps occur, from 9 to 54; the level share is 17.41 %; L = 9 holds 55 % 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.