In number theory, a -hyperperfect number is a natural number holds, where is the divisor function (i.e., the sum of all positive divisors of ). A hyperperfect number is a -hyperperfect number for some integer . Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.

The first few numbers in the sequence of -hyperperfect numbers are , with the corresponding values of being . The first few -hyperperfect numbers that are not perfect are .

List of hyperperfect numbers

The following table lists the first few -hyperperfect numbers for some values of , together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of -hyperperfect numbers:It can be shown that if is an odd integer are prime numbers, then is -hyperperfect; Judson S. McCranie has conjectured in 2000 that all -hyperperfect numbers for odd are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if are odd primes and then is -hyperperfect.

It is also possible to show that if is prime, then for all is -hyperperfect. The following table lists known values of and corresponding values of for which is -hyperperfect:

References

Further reading

Articles

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Books

• Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, (p.114-134)

External links

• MathWorld: Hyperperfect number

• A long list of hyperperfect numbers under Data