decompwlj 3D

Numbers with no 1's in their base 4 expansion

A023709 on the OEIS · family digit rule

Weight–level plate of Numbers with no 1's in their base 4 expansion
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 viewerA023709 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,988
Level class, k > L13,804 · 13.81 %
Weight class, k ≤ L86,184 · 86.19 %
Ties, k = L21
On the level line L = 14,893
Forced level, l ≤ d²180
Range of a(n)0 … 2,896,560
Range of the jump d1 … 1,048,577
Largest weight k, level L2,896,393, 1,448,279

12 different gaps occur, from 1 to 1,048,577; the level share is 13.81 %; L = 2 holds 56 % of the level class; 12 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.