decompwlj 3D

Alternate number system in base 4

A084544 on the OEIS · family digit rule

Weight–level plate of Alternate number system 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 viewerA084544 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,989
Level class, k > L11,163 · 11.16 %
Weight class, k ≤ L88,826 · 88.84 %
Ties, k = L1
On the level line L = 17,478
Forced level, l ≤ d²428
Range of a(n)1 … 114,122,134
Range of the jump d1 … 66,666,667
Largest weight k, level L114,122,131, 57,061,066

9 different gaps occur, from 1 to 66,666,667; the level share is 11.16 %; L = 1 holds 67 % of the level class; 11 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.