decompwlj 3D

Sophie Germain primes p: 2p+1 is also prime

A005384 on the OEIS · family primes · also known as Sophie Germain primes

Weight–level plate of Sophie Germain primes p: 2p+1 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 viewerA005384 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L47,248 · 47.25 %
Weight class, k ≤ L52,747 · 52.75 %
Ties, k = L0
On the level line L = 117,109
Forced level, l ≤ d²213
Range of a(n)2 … 19,391,363
Range of the jump d1 … 2,118
Largest weight k, level L19,389,107, 2,768,231

p with 2p + 1 prime. Past 3 every term is 5 mod 6, so l = 5 (mod 6) and no tie can occur, as for the lesser twin primes. The level share, 47.25 %, is close to theirs (47.77 %).

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.