decompwlj 3D

Numbers n such that sum of digits = number of digits

A061384 on the OEIS · family digit rule

Weight–level plate of Numbers n such that sum of digits = number of digits
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 viewerA061384 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,991
Level class, k > L23,767 · 23.77 %
Weight class, k ≤ L76,224 · 76.23 %
Ties, k = L0
On the level line L = 14,674
Forced level, l ≤ d²3,016
Range of a(n)1 … 10,205,000,102
Range of the jump d9 … 900,000,019
Largest weight k, level L10,204,001,993, 1,010,000,000

88 different gaps occur, from 9 to 900,000,019; the level share is 23.77 %; 3.0 % of terms are forced level (l <= d^2); there are no ties; 9 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.