decompwlj 3D

Beatty sequence for Pi^e

A038153 on the OEIS · family Beatty

Weight–level plate of Beatty sequence for Pi^e
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 viewerA038153 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L30,452 · 30.45 %
Weight class, k ≤ L69,546 · 69.55 %
Ties, k = L10
On the level line L = 17,433
Forced level, l ≤ d²21
Range of a(n)22 … 2,245,915
Range of the jump d22 … 23
Largest weight k, level L2,245,871, 97,629

The gaps are 22 and 23; the level share is 30.45 %.

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.