decompwlj 3D

Numbers whose sum of divisors is a square

A006532 on the OEIS · family divisor functions

Weight–level plate of Numbers whose sum of divisors is a square
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 viewerA006532 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L45,376 · 45.38 %
Weight class, k ≤ L54,621 · 54.62 %
Ties, k = L6
On the level line L = 15,464
Forced level, l ≤ d²297
Range of a(n)1 … 48,838,206
Range of the jump d1 … 6,783
Largest weight k, level L48,823,823, 24,398,434

3,139 different gaps occur, from 1 to 6,783; the level share is 45.38 %.

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.