Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n

A111592 on the OEIS · family divisor functions

Weight–level plate of Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n — 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 A111592 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,999
Level class, k > L	24,847 · 24.85 %
Weight class, k ≤ L	75,152 · 75.15 %
Ties, k = L	0
On the level line L = 1	7
Forced level, l ≤ d²	52
Range of a(n)	12 … 6,039,078
Range of the jump d	2 … 516
Largest weight k, level L	3,019,529, 1,509,010

179 different gaps occur, from 2 to 516; the level share is 24.85 %; there are no ties.

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.