decompwlj 3D

Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2

A002411 on the OEIS · family polynomial · also known as Pentagonal pyramidal numbers

Weight–level plate of Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2
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 viewerA002411 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,993
Level class, k > L99,993 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 11,252
Forced level, l ≤ d²99,993
Range of a(n)0 … 499,990,000,050,000
Range of the jump d1 … 14,999,950,000
Largest weight k, level L497,474,264,551,379, 33,231

n^2 (n + 1)/2, the pentagonal pyramidal numbers. d grows like n^2 while l grows like n^3, so l/d^2 -> 0: every decomposable term is forced level (100 %).

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.