decompwlj 3D

Primes of the form k^2 + k + 41

A005846 on the OEIS · family primes

Weight–level plate of Primes of the form k^2 + k + 41
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 viewerA005846 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)100,000
Level class, k > L99,999 · 100.00 %
Weight class, k ≤ L1 · 0.00 %
Ties, k = L0
On the level line L = 17,295
Forced level, l ≤ d²99,997
Range of a(n)41 … 122,978,917,213
Range of the jump d2 … 22,664,950
Largest weight k, level L122,974,709,029, 174,497

96,902 different gaps occur, from 2 to 22,664,950; every decomposable term is forced level (l <= d^2).

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.