decompwlj 3D

Primes of the form 4x^2+4xy+5y^2, with x and y nonnegative

A106931 on the OEIS · family quadratic form

Weight–level plate of Primes of the form 4x^2+4xy+5y^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 viewerA106931 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L35,850 · 35.85 %
Weight class, k ≤ L64,146 · 64.15 %
Ties, k = L0
On the level line L = 17,118
Forced level, l ≤ d²100
Range of a(n)5 … 8,432,797
Range of the jump d8 … 1,024
Largest weight k, level L8,429,693, 936,877

92 different gaps occur, from 8 to 1,024; the level share is 35.85 %; 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.