decompwlj 3D

Numbers k such that 5*k^2 + 1 is prime

A111147 on the OEIS · family prime values

Weight–level plate of Numbers k such that 5*k^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 viewerA111147 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L20,720 · 20.72 %
Weight class, k ≤ L79,277 · 79.28 %
Ties, k = L0
On the level line L = 10
Forced level, l ≤ d²48
Range of a(n)6 … 4,566,972
Range of the jump d6 … 450
Largest weight k, level L761,009, 570,543

70 different gaps occur, from 6 to 450; the level share is 20.72 %; L = 6 holds 40 % of the level class; there are no ties.

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.