In mathematics, a k-hyperperfect number (sometimes just called hyperperfect number) is a natural numbern for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function (i.e., the sum of all positive divisors of n). A number is perfectiff it is 1-hyperperfect.
The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, ... (sequence A034897 in OEIS), with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 in OEIS). The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... (sequence A007592 in OEIS).
The following table lists the first few k-hyperperfect numbers for some values of k, together with the sequence number in OEIS of the sequence of k-hyperperfect numbers:
It can be shown that if k > 1 is an oddinteger and p = (3k + 1) / 2 and q = 3k + 4 are prime numbers, then p²q is k-hyperperfect; Judson S. McCraine has conjectured in 2000 that all k-hyperperfect numbers for odd k > 1 are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if p ≠ q are odd primes and k is an integer such that k(p + q) = pq - 1, then pq is k-hyperperfect.
It is also possible to show that if k > 0 and p = k + 1 is prime, then for all i > 1 such that q = pi − p + 1 is prime, n = pi − 1q is k-hyperperfect. The following table lists known values of k and corresponding values of i for which n is k-hyperperfect:
* Daniel Minoli, Robert Bear, Hyperperfect Numbers, PME (Pi Mu Epsilon) Journal, University Oklahoma, Fall 1975, pp. 153-157. * Daniel Minoli, Sufficient Forms For Generalized Perfect Numbers, Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem; Vol. 4, No. 2, Dec 1978, pp. 277-302. * Daniel Minoli, Structural Issues For Hyperperfect Numbers, Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14. * Daniel Minoli, Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation, Vol. 34, No. 150, April 1980, pp. 639-645. * Daniel Minoli, New Results For Hyperperfect Numbers, Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, pp. 561. * Daniel Minoli, W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing. * Judson S. McCranie, A Study of Hyperperfect Numbers, Journal of Integer Sequences, Vol. 3 (2000), http://www.math.uwaterloo.ca/JIS/VOL3/mccranie.html
Books
* Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0071406158 (p.114-134)
