decompwlj 3D

Primes p such that p + 8 is also prime

A023202 on the OEIS · family primes · also known as Primes p with p + 8 prime

Weight–level plate of Primes p such that p + 8 is also 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 viewerA023202 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L46,999 · 47.00 %
Weight class, k ≤ L52,997 · 53.00 %
Ties, k = L0
On the level line L = 117,163
Forced level, l ≤ d²179
Range of a(n)3 … 18,468,251
Range of the jump d2 … 2,004
Largest weight k, level L18,465,773, 1,672,387

Primes p with p + 8 prime. Past 3 every term is 5 mod 6 (p + 8 must avoid 3), so l = 5 (mod 6): no ties, by proof. The level share is 47.00 %; L = 1 holds 37 % 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.