decompwlj 3D

Beatty sequence for e^Pi

A038152 on the OEIS · family Beatty

Weight–level plate of Beatty sequence for e^Pi
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 viewerA038152 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L30,705 · 30.71 %
Weight class, k ≤ L69,293 · 69.29 %
Ties, k = L13
On the level line L = 17,347
Forced level, l ≤ d²21
Range of a(n)23 … 2,314,069
Range of the jump d23 … 24
Largest weight k, level L2,313,907, 96,407

The gaps are 23 and 24; the level share is 30.71 %.

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.