Hemiperfect number

In number theory, a hemiperfect number is a positive integer with a half-integral abundancy index.

For a given odd number k, a number n is called k-hemiperfect if and only if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to k/2 × n.
Smallest k-hemiperfect numbers

The following table gives an overview of the smallest k-hemiperfect numbers for k ≤ 17 (sequence A088912 in the OEIS):

k	Smallest k-hemiperfect number	Number of digits
3	2	1
5	24	2
7	4320	4
9	8910720	7
11	17116004505600	14
13	170974031122008628879954060917200710847692800	45
15	12749472205565550032020636281352368036406720997031277595140988449695952806020854579200000^[1]	89
17	27172904004644864174776390325441204588387876949911859015099963347683477337589882757168182488651338324482275518065870009252589097916253652597707421065171952334010184222064839170719744000000000^[1]	191

For example, 24 is 5-hemiperfect because the sum of the divisors of 24 is

1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = 5/2 × 24.