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:
24 is a hemiperfect number because the sum of the divisors of 24 is
The abundancy index is 5/2 which is a half-integer.
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|
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.