decompwlj 3D

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

A106875 on the OEIS · family quadratic form

Weight–level plate of Primes of the form 3x^2+2xy+3y^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 viewerA106875 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L40,236 · 40.24 %
Weight class, k ≤ L59,760 · 59.76 %
Ties, k = L0
On the level line L = 17,514
Forced level, l ≤ d²234
Range of a(n)3 … 15,821,467
Range of the jump d8 … 1,760
Largest weight k, level L15,819,899, 1,755,035

161 different gaps occur, from 8 to 1,760; the level share is 40.24 %; 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.