decompwlj 3D

Numbers that are not the sum of 2 squares

A022544 on the OEIS · family quadratic form · also known as Not sums of two squares

Weight–level plate of Numbers that are not the sum of 2 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 viewerA022544 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L13,377 · 13.38 %
Weight class, k ≤ L86,621 · 86.62 %
Ties, k = L0
On the level line L = 110,560
Forced level, l ≤ d²1
Range of a(n)3 … 131,061
Range of the jump d1 … 4
Largest weight k, level L131,023, 65,530

Not a sum of two squares: some prime 3 mod 4 divides n to an odd power. The level share is 13.38 %; L = 1 holds 79 % of the level class; 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.