decompwlj 3D

Numbers that are the sum of 2 nonzero squares

A000404 on the OEIS · family quadratic form · also known as Sums of two nonzero squares

Weight–level plate of Numbers that are the sum of 2 nonzero squares
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 viewerA000404 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L15,969 · 15.97 %
Weight class, k ≤ L84,029 · 84.03 %
Ties, k = L17
On the level line L = 16,157
Forced level, l ≤ d²3
Range of a(n)2 … 448,082
Range of the jump d1 … 34
Largest weight k, level L448,027, 224,000

x^2 + y^2 with x, y >= 1. The plane is almost exactly that of all sums of two squares (15.97 % level against 15.95 %).

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.