decompwlj 3D

Numbers n such that sigma(n) is a nontrivial power, i.e., sigma(n) = a^b where a and b are greater than 1

A065496 on the OEIS · family divisor functions

Weight–level plate of Numbers n such that sigma(n) is a nontrivial power, i.e., sigma(n) = a^b where a and b are greater than 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 viewerA065496 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L44,305 · 44.31 %
Weight class, k ≤ L55,692 · 55.69 %
Ties, k = L5
On the level line L = 15,555
Forced level, l ≤ d²201
Range of a(n)3 … 41,598,714
Range of the jump d1 … 5,394
Largest weight k, level L41,579,107, 20,601,466

2,782 different gaps occur, from 1 to 5,394; the level share is 44.31 %.

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.