decompwlj 3D

Beatty sequence for zeta(3)/(zeta(3)-1)

A059538 on the OEIS · family Beatty

Weight–level plate of Beatty sequence for zeta(3)/(zeta(3)-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 viewerA059538 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L21,311 · 21.31 %
Weight class, k ≤ L78,687 · 78.69 %
Ties, k = L33
On the level line L = 18,255
Forced level, l ≤ d²5
Range of a(n)5 … 594,910
Range of the jump d5 … 6
Largest weight k, level L594,773, 99,133

The gaps are 5 and 6; the level share is 21.31 %; L = 1 holds 39 % 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.