decompwlj 3D

Numbers k such that (k^2 + 1)/2 is prime

A002731 on the OEIS · family prime values

Weight–level plate of Numbers k such that (k^2 + 1)/2 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 viewerA002731 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L33,028 · 33.03 %
Weight class, k ≤ L66,970 · 66.97 %
Ties, k = L31
On the level line L = 117,159
Forced level, l ≤ d²17
Range of a(n)3 … 1,898,329
Range of the jump d2 … 210
Largest weight k, level L1,898,107, 631,929

81 different gaps occur, from 2 to 210; the level share is 33.03 %; L = 1 holds 52 % of the level class.

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.