decompwlj 3D

Numbers that contain only the digits 0,4,8

A343810 on the OEIS · family digit rule

Weight–level plate of Numbers that contain only the digits 0,4,8
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 viewerA343810 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,979
Level class, k > L10,786 · 10.79 %
Weight class, k ≤ L89,193 · 89.21 %
Ties, k = L1
On the level line L = 10
Forced level, l ≤ d²632
Range of a(n)0 … 48,008,044,800
Range of the jump d4 … 31,111,111,112
Largest weight k, level L12,002,011,199, 9,601,608,896

11 different gaps occur, from 4 to 31,111,111,112; the level share is 10.79 %; L = 4 holds 36 % of the level class; 21 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.