number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

, a hemiperfect number is a

positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

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

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

s of ''n''. The first few hemiperfect numbers are: :2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960, ...

Example

24 is a hemiperfect number because the sum of the divisors of 24 is : 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = × 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 : 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 ≈ , and the smallest known number of abundancy 17/2 is ≈ . There are no known numbers of abundancy 19/2.

References

{{Classes of natural numbers Integer sequences Perfect numbers

Example

Smallest hemiperfect numbers of abundancy ''k''/2

See also

References