decompwlj 3D

"9ish numbers": decimal representation contains at least one nine

A011539 on the OEIS · family digit rule · also known as Numbers with a digit 9

Weight–level plate of "9ish numbers": decimal representation contains at least one nine
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 viewerA011539 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L13,623 · 13.62 %
Weight class, k ≤ L86,375 · 86.38 %
Ties, k = L67
On the level line L = 111,043
Forced level, l ≤ d²8
Range of a(n)9 … 253,494
Range of the jump d1 … 10
Largest weight k, level L253,493, 126,746

At least one decimal digit 9. Only the gaps 1 to 10 occur. The level share is 13.62 %; L = 1 holds 81 % 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.