decompwlj 3D

a(n) = n-th prime + n

A014688 on the OEIS · family primes · also known as n-th prime plus n

Weight–level plate of a(n) = n-th prime + n
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 viewerA014688 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L25,349 · 25.35 %
Weight class, k ≤ L74,649 · 74.65 %
Ties, k = L15
On the level line L = 17,735
Forced level, l ≤ d²7
Range of a(n)3 … 1,399,709
Range of the jump d2 … 115
Largest weight k, level L1,399,621, 349,854

p(n) + n. The gap is the prime gap plus 1. The level share is 25.35 %; L = 1 holds 31 % 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.