decompwlj 3D

Primes of the form 4x^2 + 13y^2

A107159 on the OEIS · family quadratic form

Weight–level plate of Primes of the form 4x^2 + 13y^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 viewerA107159 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L39,979 · 39.98 %
Weight class, k ≤ L60,019 · 60.02 %
Ties, k = L14
On the level line L = 18,456
Forced level, l ≤ d²163
Range of a(n)13 … 12,195,637
Range of the jump d4 … 1,332
Largest weight k, level L12,194,977, 2,424,561

238 different gaps occur, from 4 to 1,332; the level share is 39.98 %.

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.