decompwlj 3D

Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice

A003136 on the OEIS · family quadratic form · also known as Loeschian numbers

Weight–level plate of Loeschian numbers: numbers of the form x^2 + xy + y^2; norms of vectors in A2 lattice
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 viewerA003136 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L24,232 · 24.23 %
Weight class, k ≤ L75,765 · 75.77 %
Ties, k = L69
On the level line L = 115,684
Forced level, l ≤ d²1
Range of a(n)0 … 538,689
Range of the jump d1 … 39
Largest weight k, level L538,651, 269,281

x^2 + xy + y^2: every prime 2 mod 3 divides n to an even power. Starts at a(1) = 0. The level share, 24.23 %, is well above the sums of two squares' 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.