decompwlj 3D

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

A106856 on the OEIS · family primes

Weight–level plate of Primes of the form x^2 + xy + 2y^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 viewerA106856 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L27,753 · 27.75 %
Weight class, k ≤ L72,244 · 72.25 %
Ties, k = L7
On the level line L = 15,882
Forced level, l ≤ d²42
Range of a(n)2 … 3,649,417
Range of the jump d2 … 378
Largest weight k, level L3,648,427, 1,216,313

142 different gaps occur, from 2 to 378; the level share is 27.75 %.

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.