decompwlj 3D

Twin primes

A001097 on the OEIS · family primes

Weight–level plate of Twin 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 viewerA001097 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L19,672 · 19.67 %
Weight class, k ≤ L80,325 · 80.33 %
Ties, k = L4
On the level line L = 16
Forced level, l ≤ d²178
Range of a(n)3 … 8,265,137
Range of the jump d2 … 1,528
Largest weight k, level L2,752,879, 2,755,045

Primes in a twin pair, both members. Past 5 every l is 3 mod 6: after a lesser twin d = 2 and l = p - 2; after a greater twin d = 4 (mod 6). So 3 | l, and the line L = 1 holds only 6 terms. All but 68 level terms sit on odd multiples of 3 (L = 3, 9, 15, ...). The level share is 19.67 %.

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.