decompwlj 3D

Numbers of the form 2^k + prime

A118955 on the OEIS · family primes

Weight–level plate of Numbers of the form 2^k + prime
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 viewerA118955 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)100,000
Level class, k > L32,156 · 32.16 %
Weight class, k ≤ L67,844 · 67.84 %
Ties, k = L84
On the level line L = 131,876
Forced level, l ≤ d²0
Range of a(n)3 … 179,977
Range of the jump d1 … 10
Largest weight k, level L179,969, 89,984

9 different gaps occur, from 1 to 10; the level share is 32.16 %; L = 1 holds 99 % 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.