decompwlj 3D

a(n) = n-th prime + n-th nonprime

A022797 on the OEIS · family primes

Weight–level plate of a(n) = n-th prime + n-th nonprime
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 viewerA022797 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L26,115 · 26.12 %
Weight class, k ≤ L73,883 · 73.88 %
Ties, k = L20
On the level line L = 18,944
Forced level, l ≤ d²6
Range of a(n)3 … 1,410,195
Range of the jump d3 … 115
Largest weight k, level L1,409,531, 352,475

94 different gaps occur, from 3 to 115; the level share is 26.12 %; L = 1 holds 34 % 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.