decompwlj 3D

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

A127435 on the OEIS · family primes

Weight–level plate of Primes p such that (p-1)^2 + 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 viewerA127435 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,993
Level class, k > L42,790 · 42.79 %
Weight class, k ≤ L57,203 · 57.21 %
Ties, k = L3
On the level line L = 17,718
Forced level, l ≤ d²256
Range of a(n)2 … 22,689,691
Range of the jump d1 … 2,480
Largest weight k, level L22,686,527, 3,237,129

515 different gaps occur, from 1 to 2,480; the level share is 42.79 %; 7 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.