decompwlj 3D

Numbers whose sum of digits is 9

A052223 on the OEIS · family digit rule

Weight–level plate of Numbers whose sum of digits is 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 viewerA052223 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L13,041 · 13.04 %
Weight class, k ≤ L86,957 · 86.96 %
Ties, k = L2
On the level line L = 12
Forced level, l ≤ d²3,967
Range of a(n)9 … 100,101,220,101
Range of the jump d9 … 10,000,000,008
Largest weight k, level L21,333,333,333, 8,341,768,341

83 different gaps occur, from 9 to 10,000,000,008; the level share is 13.04 %; 4.0 % of terms are forced level (l <= d^2); L = 9 holds 33 % 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.