decompwlj 3D

Primes of form x^2+67*y^2

A033243 on the OEIS · family quadratic form

Weight–level plate of Primes of form x^2+67*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 viewerA033243 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)100,000
Level class, k > L37,050 · 37.05 %
Weight class, k ≤ L62,950 · 62.95 %
Ties, k = L8
On the level line L = 18,432
Forced level, l ≤ d²135
Range of a(n)67 … 8,974,597
Range of the jump d2 … 1,118
Largest weight k, level L8,970,217, 2,987,693

347 different gaps occur, from 2 to 1,118; the level share is 37.05 %.

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.