decompwlj 3D

Number of primes less than 1000n

A038812 on the OEIS · family primes

Weight–level plate of Number of primes less than 1000n
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 viewerA038812 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L36,733 · 36.73 %
Weight class, k ≤ L63,266 · 63.27 %
Ties, k = L4
On the level line L = 16,932
Forced level, l ≤ d²77
Range of a(n)168 … 5,761,455
Range of the jump d34 … 135
Largest weight k, level L5,758,723, 141,527

84 different gaps occur, from 34 to 135; the level share is 36.73 %.

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.