decompwlj 3D

Numbers of form 4 x^2 + 5 y^2

A020682 on the OEIS · family quadratic form

Weight–level plate of Numbers of form 4 x^2 + 5 y^2
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 viewerA020682 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L27,736 · 27.74 %
Weight class, k ≤ L72,260 · 72.26 %
Ties, k = L33
On the level line L = 111,572
Forced level, l ≤ d²11
Range of a(n)0 … 1,009,780
Range of the jump d1 … 87
Largest weight k, level L1,009,669, 336,581

55 different gaps occur, from 1 to 87; the level share is 27.74 %; L = 1 holds 42 % of the level class.

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.