decompwlj 3D

Numbers having exactly one representation by the quadratic form x^2 + xy + y^2 with 0 <= x <= y

A198772 on the OEIS · family quadratic form

Weight–level plate of Numbers having exactly one representation by the quadratic form x^2 + xy + y^2 with 0 <= x <= y
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 viewerA198772 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L27,062 · 27.06 %
Weight class, k ≤ L72,935 · 72.94 %
Ties, k = L25
On the level line L = 113,298
Forced level, l ≤ d²2
Range of a(n)0 … 1,055,884
Range of the jump d1 … 107
Largest weight k, level L1,055,827, 527,881

84 different gaps occur, from 1 to 107; the level share is 27.06 %; L = 1 holds 49 % of the level class.

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.