Hemiperfect number

Innumber theory,ahemiperfect numberis apositive integerwith a half-integerabundancy index.In other words,σ(n)/n=k/2 for an odd integerk,whereσ(n) is thesum-of-divisors function,the sum of all positivedivisorsofn.

The first few hemiperfect numbers are:

2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960,... (sequenceA159907in theOEIS)

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 abundancyk/2

The following table gives an overview of the smallest hemiperfect numbers of abundancyk/2 fork≤ 13 (sequenceA088912in theOEIS):

k	Smallest number of abundancyk/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.

See also

• Semiperfect number
• Perfect number
• Multiply perfect number