# 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 divisor 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, ... (sequence A159907 in the OEIS)

## 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 (sequence A088912 in the OEIS):

k Smallest number of abundancy k/2 Number of digits
3 2 1
5 24 2
7 4320 4
9 8910720 7
11 17116004505600 14
13 170974031122008628879954060917200710847692800 45

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 ≈ 1.274947×1088, and the smallest known number of abundancy 17/2 is ≈ 2.717290×10190.

There are no known numbers of abundancy 19/2.