Hemiperfect number

In number theory, a hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the sum-of-divisors function, the sum of all positive divisors of n.

The first few hemiperfect numbers are:


 * 2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960, ...

Example
24 is a hemiperfect number because the sum of the divisors of 24 is


 * 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = $5⁄2$ × 24.

The abundancy index is 5/2 which is a half-integer.

Smallest hemiperfect numbers of abundancy k/2
The following table gives an overview of the smallest hemiperfect numbers of abundancy k/2 for k ≤ 13 :

The current best known upper bounds for the smallest numbers of abundancy 15/2 and 17/2 were found by Michel Marcus.

The smallest known number of abundancy 15/2 is ≈ $4,320$, and the smallest known number of abundancy 17/2 is ≈ $8,910,720$.

There are no known numbers of abundancy 19/2.