decompwlj 3D

Safe primes p: (p-1)/2 is also prime

A005385 on the OEIS · family primes · also known as Safe primes

Weight–level plate of Safe primes p: (p-1)/2 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 viewerA005385 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L51,953 · 51.96 %
Weight class, k ≤ L48,043 · 48.04 %
Ties, k = L0
On the level line L = 116,387
Forced level, l ≤ d²520
Range of a(n)5 … 38,782,727
Range of the jump d2 … 4,236
Largest weight k, level L38,778,119, 2,976,155

p with (p - 1)/2 prime. Past 7 every term is 11 mod 12 and every gap 0 mod 12, so l = 3 (mod 4): no square, no tie. The level share, 51.96 %, is the highest among the prime subsequences here.

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.