Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers

A005153 on the OEIS · family divisor functions · also known as Practical numbers

Weight–level plate of Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers — 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 viewer A005153 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,996
Level class, k > L	17,503 · 17.50 %
Weight class, k ≤ L	82,493 · 82.50 %
Ties, k = L	1
On the level line L = 1	2
Forced level, l ≤ d²	4
Range of a(n)	1 … 1,028,952
Range of the jump d	1 … 64
Largest weight k, level L	513,841, 257,113

Practical numbers. Every term > 2 is divisible by 4 or 6, so a and d are even and 2 | l at every decomposable term: the line L = 1 holds only 2 terms and the level class sits on L = 4, 6, 8, ... The level share is 17.50 %.

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.