decompwlj 3D

Numbers in base 4

A007090 on the OEIS · family digit rule · also known as Base 4 read in decimal

Weight–level plate of Numbers in base 4
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 viewerA007090 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,982
Level class, k > L12,648 · 12.65 %
Weight class, k ≤ L87,334 · 87.35 %
Ties, k = L9
On the level line L = 17,417
Forced level, l ≤ d²337
Range of a(n)0 … 120,122,133
Range of the jump d1 … 66,666,667
Largest weight k, level L120,122,111, 60,061,065

n in base 4, read in decimal. The gaps are 1, 7, 67, 667, ... The level share is 12.65 %; L = 1 holds 59 % 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.