decompwlj 3D

Sums of two nonnegative cubes

A004999 on the OEIS · family quadratic form

Weight–level plate of Sums of two nonnegative cubes
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 viewerA004999 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,994
Level class, k > L49,923 · 49.93 %
Weight class, k ≤ L50,071 · 50.07 %
Ties, k = L3
On the level line L = 15,484
Forced level, l ≤ d²1,266
Range of a(n)0 … 107,865,801
Range of the jump d1 … 16,955
Largest weight k, level L107,842,789, 53,585,937

5,360 different gaps occur, from 1 to 16,955; the level share is 49.93 %; 1.3 % of terms are forced level (l <= d^2); 6 terms do not decompose.

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.