decompwlj 3D

Primes p such that p+6 and p+8 are also primes

A046138 on the OEIS · family primes

Weight–level plate of Primes p such that p+6 and p+8 are also primes
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 viewerA046138 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L63,704 · 63.71 %
Weight class, k ≤ L36,291 · 36.29 %
Ties, k = L0
On the level line L = 115,579
Forced level, l ≤ d²3,914
Range of a(n)5 … 205,216,721
Range of the jump d6 … 23,310
Largest weight k, level L205,145,243, 15,265,643

1,397 different gaps occur, from 6 to 23,310; the level share is 63.71 %; 3.9 % 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.