decompwlj 3D

Numbers of the form x^2 + 2*y^2

A002479 on the OEIS · family quadratic form · also known as Numbers x^2 + 2y^2

Weight–level plate of Numbers of the form x^2 + 2*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 viewerA002479 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L14,910 · 14.91 %
Weight class, k ≤ L85,086 · 85.09 %
Ties, k = L27
On the level line L = 17,430
Forced level, l ≤ d²3
Range of a(n)0 … 387,857
Range of the jump d1 … 37
Largest weight k, level L387,781, 193,880

x^2 + 2y^2: every prime 5 or 7 mod 8 divides n to an even power. Starts at a(1) = 0. Level share 14.91 %, with half the level class on L = 1.

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.