decompwlj 3D

Numbers that are sums of consecutive squares

A034705 on the OEIS · family summatory

Weight–level plate of Numbers that are sums of consecutive 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 viewerA034705 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L38,529 · 38.53 %
Weight class, k ≤ L61,467 · 61.47 %
Ties, k = L19
On the level line L = 16,602
Forced level, l ≤ d²25
Range of a(n)0 … 13,948,689
Range of the jump d1 … 2,031
Largest weight k, level L13,944,401, 6,971,377

1,128 different gaps occur, from 1 to 2,031; the level share is 38.53 %.

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.