decompwlj 3D

Beatty sequence for Pi^2/6, or zeta(2)

A059535 on the OEIS · family Beatty

Weight–level plate of Beatty sequence for Pi^2/6, or zeta(2)
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 viewerA059535 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L12,370 · 12.37 %
Weight class, k ≤ L87,628 · 87.63 %
Ties, k = L52
On the level line L = 19,225
Forced level, l ≤ d²1
Range of a(n)1 … 164,493
Range of the jump d1 … 2
Largest weight k, level L164,471, 82,242

The gaps are 1 and 2; the level share is 12.37 %; L = 1 holds 75 % 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.