decompwlj 3D

Primes of the form 3x^2+xy+4y^2, with x and y nonnegative

A106894 on the OEIS · family quadratic form

Weight–level plate of Primes of the form 3x^2+xy+4y^2, with x and y nonnegative
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 viewerA106894 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L39,860 · 39.86 %
Weight class, k ≤ L60,138 · 60.14 %
Ties, k = L9
On the level line L = 17,817
Forced level, l ≤ d²285
Range of a(n)3 … 17,171,129
Range of the jump d2 … 2,330
Largest weight k, level L17,166,683, 5,722,495

639 different gaps occur, from 2 to 2,330; the level share is 39.86 %.

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.