decompwlj 3D

Numbers whose ternary expansion contains no 0

A032924 on the OEIS · family digit rule · also known as No digit 0 in base 3

Weight–level plate of Numbers whose ternary expansion contains no 0
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 viewerA032924 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,984
Level class, k > L4,862 · 4.86 %
Weight class, k ≤ L95,122 · 95.14 %
Ties, k = L0
On the level line L = 19
Forced level, l ≤ d²372
Range of a(n)1 … 35,953,430
Range of the jump d1 … 7,174,454
Largest weight k, level L21,523,359, 17,976,714

No digit 0 in base 3: the digits are 1 and 2. After the numbers not divisible by 3 it has the lowest level share in the atlas, and the line L = 1 holds only 9 terms. The level share is 4.86 %; L = 3 holds 53 % of the level class; there are no ties.

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.