decompwlj 3D

Numbers with no 3's in base-4 expansion

A023717 on the OEIS · family digit rule

Weight–level plate of Numbers with no 3's in 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 viewerA023717 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,988
Level class, k > L7,808 · 7.81 %
Weight class, k ≤ L92,180 · 92.19 %
Ties, k = L2
On the level line L = 15,139
Forced level, l ≤ d²183
Range of a(n)0 … 1,581,408
Range of the jump d1 … 349,526
Largest weight k, level L1,581,379, 790,700

11 different gaps occur, from 1 to 349,526; the level share is 7.81 %; L = 1 holds 66 % 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.