decompwlj 3D

Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two

A122535 on the OEIS · family primes

Weight–level plate of Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two
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 viewerA122535 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L46,533 · 46.53 %
Weight class, k ≤ L53,464 · 53.47 %
Ties, k = L0
On the level line L = 16,984
Forced level, l ≤ d²1,000
Range of a(n)3 … 56,206,673
Range of the jump d6 … 6,940
Largest weight k, level L56,185,627, 8,013,325

1,838 different gaps occur, from 6 to 6,940; the level share is 46.53 %; 1.0 % of terms are forced level (l <= d^2); there are no ties.

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.