decompwlj 3D

Numbers k with the property that the sum of the digits of k is a substring of k

A052018 on the OEIS · family digit rule

Weight–level plate of Numbers k with the property that the sum of the digits of k is a substring of k
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 viewerA052018 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,994
Level class, k > L20,512 · 20.51 %
Weight class, k ≤ L79,482 · 79.49 %
Ties, k = L4
On the level line L = 17,024
Forced level, l ≤ d²208
Range of a(n)0 … 1,701,161
Range of the jump d1 … 882
Largest weight k, level L1,701,137, 850,508

292 different gaps occur, from 1 to 882; the level share is 20.51 %; L = 1 holds 34 % of the level class; 6 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.