decompwlj 3D

Balanced primes (of order one): primes which are the average of the previous prime and the following prime

A006562 on the OEIS · family primes · also known as Balanced primes

Weight–level plate of Balanced primes (of order one): primes which are the average of the previous prime and the following 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 viewerA006562 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L52,419 · 52.42 %
Weight class, k ≤ L47,578 · 47.58 %
Ties, k = L3
On the level line L = 113,471
Forced level, l ≤ d²1,001
Range of a(n)5 … 56,206,697
Range of the jump d6 … 6,970
Largest weight k, level L56,206,649, 3,306,093

Primes that are the mean of their two neighbours. They are sparse (the 100,000th is 56,206,697), and the gaps are large and varied. The level share is 52.42 %; 1.0 % of terms are forced level (l <= d^2).

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.