decompwlj 3D

Initial members of prime triples (p, p+2, p+6)

A022004 on the OEIS · family primes · also known as Prime triplets (p, p+2, p+6)

Weight–level plate of Initial members of prime triples (p, p+2, p+6)
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 viewerA022004 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L63,420 · 63.42 %
Weight class, k ≤ L36,575 · 36.58 %
Ties, k = L0
On the level line L = 115,492
Forced level, l ≤ d²3,853
Range of a(n)5 … 203,855,081
Range of the jump d6 … 25,746
Largest weight k, level L203,849,027, 18,041,215

p with p, p + 2, p + 6 all prime. Past 5 every term is 5 mod 6, so l = 5 (mod 6): no ties, by proof. The level share is 63.42 %; 3.9 % 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.