decompwlj 3D

The primes doubled; even semiprimes

A100484 on the OEIS · family primes · also known as Twice the primes

Weight–level plate of The primes doubled; even semiprimes
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 viewerA100484 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L23,004 · 23.00 %
Weight class, k ≤ L76,993 · 77.00 %
Ties, k = L0
On the level line L = 11
Forced level, l ≤ d²19
Range of a(n)4 … 2,599,418
Range of the jump d2 … 228
Largest weight k, level L1,299,061, 519,774

2p. The primes' decomposition doubled: d = 2g and l = 2(p - g), so the primes' level line L = 1 moves to L = 2. 23,004 terms are level-classified, against 22,999 for the primes. The level share is 23.00 %; L = 2 holds 32 % of the level class; 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.