decompwlj 3D

Abundant numbers (sum of divisors of m exceeds 2m)

A005101 on the OEIS · family divisor functions · also known as Abundant numbers

Weight–level plate of Abundant numbers (sum of divisors of m exceeds 2m)
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 viewerA005101 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L14,898 · 14.90 %
Weight class, k ≤ L85,101 · 85.10 %
Ties, k = L1
On the level line L = 132
Forced level, l ≤ d²4
Range of a(n)12 … 404,064
Range of the jump d1 … 6
Largest weight k, level L394,063, 196,927

sigma(n) > 2n. 99.2 % of these terms are even (the first odd one is 945), and so is l on 99.2 % of decomposable terms: the level class sits on even L, and the line L = 1 holds only 32 terms.

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.