decompwlj 3D

Numbers that are a product of 2 triangular numbers

A085780 on the OEIS · family polynomial

Weight–level plate of Numbers that are a product of 2 triangular numbers
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 viewerA085780 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L42,168 · 42.17 %
Weight class, k ≤ L57,827 · 57.83 %
Ties, k = L1
On the level line L = 1960
Forced level, l ≤ d²760
Range of a(n)0 … 125,167,770
Range of the jump d1 … 15,457
Largest weight k, level L125,091,467, 62,027,437

5,749 different gaps occur, from 1 to 15,457; the level share is 42.17 %.

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.