decompwlj 3D

Primes of the form x^2 + 165*y^2

A139648 on the OEIS · family quadratic form

Weight–level plate of Primes of the form x^2 + 165*y^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 viewerA139648 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L53,554 · 53.56 %
Weight class, k ≤ L46,444 · 46.44 %
Ties, k = L57
On the level line L = 123,669
Forced level, l ≤ d²357
Range of a(n)181 … 25,667,269
Range of the jump d12 … 2,640
Largest weight k, level L25,666,897, 1,973,929

103 different gaps occur, from 12 to 2,640; the level share is 53.56 %; L = 1 holds 44 % 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.