decompwlj 3D

Klarner-Rado sequence: a(1) = 1; subsequent terms are defined by the rule that if m is present so are 2m+1 and 3m+1

A002977 on the OEIS · family self-referential

Weight–level plate of Klarner-Rado sequence: a(1) = 1; subsequent terms are defined by the rule that if m is present so are 2m+1 and 3m+1
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 viewerA002977 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L23,553 · 23.55 %
Weight class, k ≤ L76,445 · 76.45 %
Ties, k = L13
On the level line L = 115,808
Forced level, l ≤ d²499
Range of a(n)1 … 2,911,581
Range of the jump d1 … 15,737
Largest weight k, level L2,911,481, 1,455,790

522 different gaps occur, from 1 to 15,737; the level share is 23.55 %; L = 1 holds 67 % 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.