decompwlj 3D

Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m)

A003052 on the OEIS · family digit rule · also known as Self (Colombian) numbers

Weight–level plate of Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m)
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 viewerA003052 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L24,925 · 24.93 %
Weight class, k ≤ L75,071 · 75.07 %
Ties, k = L19
On the level line L = 17,189
Forced level, l ≤ d²11
Range of a(n)1 … 1,022,675
Range of the jump d2 … 54
Largest weight k, level L1,022,653, 340,872

Self numbers: not of the form m + digitsum(m). They come in runs spaced 11 apart - gap 11 on 89.8 % of the terms - broken at decade boundaries. The level share is 24.93 %.

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.