Hyperperfect Number - List of Hyperperfect Numbers

List of Hyperperfect Numbers

The following table lists the first few k-hyperperfect numbers for some values of k, together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of k-hyperperfect numbers:

It can be shown that if k > 1 is an odd integer and p = (3k + 1) / 2 and q = 3k + 4 are prime numbers, then p²q is k-hyperperfect; Judson S. McCranie 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: