decompwlj 3D

Primes p such that 4p^2 + 1 is also prime

A052291 on the OEIS · family primes

Weight–level plate of Primes p such that 4p^2 + 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 viewerA052291 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,994
Level class, k > L45,972 · 45.97 %
Weight class, k ≤ L54,022 · 54.03 %
Ties, k = L10
On the level line L = 110,435
Forced level, l ≤ d²291
Range of a(n)2 … 23,736,443
Range of the jump d1 … 3,630
Largest weight k, level L23,731,733, 3,389,577

540 different gaps occur, from 1 to 3,630; the level share is 45.97 %; 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.