decompwlj 3D

Primes of the form x^2+xy+21y^2, with x and y nonnegative

A106969 on the OEIS · family quadratic form

Weight–level plate of Primes of the form x^2+xy+21y^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 viewerA106969 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L37,228 · 37.23 %
Weight class, k ≤ L62,770 · 62.77 %
Ties, k = L12
On the level line L = 18,268
Forced level, l ≤ d²140
Range of a(n)23 … 9,703,427
Range of the jump d2 … 1,010
Largest weight k, level L9,703,307, 3,215,179

384 different gaps occur, from 2 to 1,010; the level share is 37.23 %.

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.