decompwlj 3D

Initial members of prime triples (p, p+4, p+6)

A022005 on the OEIS · family primes · also known as Prime triplets (p, p+4, p+6)

Weight–level plate of Initial members of prime triples (p, p+4, 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 viewerA022005 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L66,201 · 66.20 %
Weight class, k ≤ L33,795 · 33.80 %
Ties, k = L12
On the level line L = 120,799
Forced level, l ≤ d²3,843
Range of a(n)7 … 203,885,677
Range of the jump d6 … 38,340
Largest weight k, level L203,870,803, 29,024,143

p with p, p + 4, p + 6 all prime. Past 7 every term is 1 mod 6. The level share is 66.20 %; 3.8 % of terms are forced level (l <= d^2); L = 1 holds 31 % 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.