decompwlj 3D

Numbers without 1 as a digit

A052383 on the OEIS · family digit rule · also known as No digit 1

Weight–level plate of Numbers without 1 as a digit
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 viewerA052383 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,993
Level class, k > L9,309 · 9.31 %
Weight class, k ≤ L90,684 · 90.69 %
Ties, k = L27
On the level line L = 17,368
Forced level, l ≤ d²24
Range of a(n)0 … 273,260
Range of the jump d1 … 100,001
Largest weight k, level L273,253, 136,629

No decimal digit 1. Gaps jump across every block with a 1 (the largest here is 100,001). The level share is 9.31 %; L = 1 holds 79 % 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.