decompwlj 3D

Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1)

A002378 on the OEIS · family polynomial · also known as Oblong numbers

Weight–level plate of Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1)
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 viewerA002378 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,995
Level class, k > L99,995 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 12
Forced level, l ≤ d²99,995
Range of a(n)0 … 9,999,900,000
Range of the jump d2 … 200,000
Largest weight k, level L4,998,750,077, 49,769

n(n + 1) = 2 T(n): d = 2n + 2 and l = (n - 2)(n + 1) < d^2, forced level (100 %). l is even and l/2 > d for n > 6, so L >= 2: the line L = 1 holds only a = 30 and 42.

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.