decompwlj 3D

Primes of the form 3x^2 + 31y^2

A107210 on the OEIS · family quadratic form

Weight–level plate of Primes of the form 3x^2 + 31y^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 viewerA107210 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L42,753 · 42.75 %
Weight class, k ≤ L57,244 · 57.25 %
Ties, k = L0
On the level line L = 115,406
Forced level, l ≤ d²151
Range of a(n)3 … 12,192,391
Range of the jump d12 … 1,320
Largest weight k, level L12,189,871, 937,363

88 different gaps occur, from 12 to 1,320; the level share is 42.75 %; L = 1 holds 36 % 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.