decompwlj 3D

Squares and twice squares

A028982 on the OEIS · family quadratic form

Weight–level plate of Squares and twice squares
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 viewerA028982 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L66,535 · 66.54 %
Weight class, k ≤ L33,461 · 33.46 %
Ties, k = L0
On the level line L = 11,917
Forced level, l ≤ d²58,582
Range of a(n)1 … 3,431,499,241
Range of the jump d1 … 117,155
Largest weight k, level L3,426,346,223, 1,119,638,520

34,684 different gaps occur, from 1 to 117,155; the level share is 66.54 %; 58.6 % of terms are forced level (l <= d^2); there are no ties.

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.