Flavius Josephus's sieve: Start with the natural numbers; at the k-th sieving step, remove every (k+1)-st term of the sequence remaining after the (k-1)-st sieving step; iterate

A000960 on the OEIS · family sieve · also known as Flavius Josephus sieve

Weight–level plate of Flavius Josephus's sieve: Start with the natural numbers; at the k-th sieving step, remove every (k+1)-st term of the sequence remaining after the (k-1)-st sieving step; iterate — 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 viewer A000960 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,997
Level class, k > L	96,561 · 96.56 %
Weight class, k ≤ L	3,436 · 3.44 %
Ties, k = L	2
On the level line L = 1	13,258
Forced level, l ≤ d²	77,425
Range of a(n)	1 … 7,854,038,953
Range of the jump d	2 … 521,092
Largest weight k, level L	7,853,264,669, 5,149,949

count(x) = 2 sqrt(x/pi), so a(n) ~ pi n^2/4 and the gap outgrows sqrt(l): l <= d^2 on 77 % of terms, and 96.6 % of the sequence is level-classified. It is the spread of the gap, not its mean, that populates the weight 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.