decompwlj 3D

Numbers that are the sum of 2 squares

A001481 on the OEIS · family multiplicative · also known as Sums of two squares

Weight–level plate of Numbers that are 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 viewerA001481 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L15,945 · 15.95 %
Weight class, k ≤ L84,051 · 84.05 %
Ties, k = L17
On the level line L = 16,154
Forced level, l ≤ d²2
Range of a(n)0 … 446,665
Range of the jump d1 … 34
Largest weight k, level L446,647, 223,312

n = x^2 + y^2: every prime 3 mod 4 divides n to an even power. Starts at a(1) = 0. Density ~ 0.764 x / sqrt(log x) (Landau-Ramanujan). The gaps stay below 35 here and the level share is 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.