decompwlj 3D

Primes p such that p^2 - p + 1 is prime

A065508 on the OEIS · family primes

Weight–level plate of Primes p such that p^2 - p + 1 is 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 viewerA065508 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,994
Level class, k > L50,262 · 50.27 %
Weight class, k ≤ L49,732 · 49.73 %
Ties, k = L18
On the level line L = 115,608
Forced level, l ≤ d²456
Range of a(n)2 … 34,697,821
Range of the jump d1 … 4,380
Largest weight k, level L34,694,887, 3,008,837

457 different gaps occur, from 1 to 4,380; the level share is 50.27 %; L = 1 holds 31 % of the level class; 6 terms do not decompose.

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.