decompwlj 3D

Beatty sequence for sqrt(Pi)

A037086 on the OEIS · family Beatty

Weight–level plate of Beatty sequence for sqrt(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 viewerA037086 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,997
Level class, k > L12,831 · 12.83 %
Weight class, k ≤ L87,166 · 87.17 %
Ties, k = L52
On the level line L = 19,025
Forced level, l ≤ d²1
Range of a(n)0 … 177,243
Range of the jump d1 … 2
Largest weight k, level L177,239, 88,615

The gaps are 1 and 2; the level share is 12.83 %; L = 1 holds 70 % 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.