decompwlj 3D

Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor

A091191 on the OEIS · family divisor functions

Weight–level plate of Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor
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 viewerA091191 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L25,309 · 25.31 %
Weight class, k ≤ L74,689 · 74.69 %
Ties, k = L1
On the level line L = 152
Forced level, l ≤ d²44
Range of a(n)12 … 5,687,562
Range of the jump d1 … 516
Largest weight k, level L5,683,213, 2,161,417

268 different gaps occur, from 1 to 516; the level share is 25.31 %.

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.