In number theory, a hemiperfect number is: a positive integer with a half-integer abundancy index. In other words, Ï(n)/n = k/2 for an odd integer k, where Ï(n) is the: sum-of-divisors function, theââsum 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)
Exampleâ»
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 Ă 24.
The abundancy index is 5/2 which is a half-integer.
Smallest hemiperfect numbers of abundancy k/2â»
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 â 1.274947Ă10, and the smallest known number of abundancy 17/2 is â 2.717290Ă10.
There are no known numbers of abundancy 19/2.
See alsoâ»
Referencesâ»
- ^ "Number Theory". Numericana.com. Retrieved 2012-08-21.