decompwlj 3D

Primes of form x^2+51*y^2

A033233 on the OEIS · family quadratic form

Weight–level plate of Primes of form x^2+51*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 viewerA033233 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L45,404 · 45.40 %
Weight class, k ≤ L54,594 · 54.60 %
Ties, k = L20
On the level line L = 115,235
Forced level, l ≤ d²282
Range of a(n)67 … 18,852,373
Range of the jump d6 … 2,244
Largest weight k, level L18,849,997, 2,685,913

249 different gaps occur, from 6 to 2,244; the level share is 45.40 %; L = 1 holds 34 % 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.