decompwlj 3D

Primes of the form 21x^2+65y^2

A140023 on the OEIS · family quadratic form

Weight–level plate of Primes of the form 21x^2+65y^2
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 viewerA140023 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L56,151 · 56.15 %
Weight class, k ≤ L43,847 · 43.85 %
Ties, k = L0
On the level line L = 118,081
Forced level, l ≤ d²867
Range of a(n)149 … 53,541,209
Range of the jump d12 … 6,912
Largest weight k, level L53,540,273, 3,125,761

201 different gaps occur, from 12 to 6,912; the level share is 56.15 %; L = 1 holds 32 % 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.