Hyperperfect number

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 = p^{i − 1}q is k-hyperperfect. The following table lists known values of k and corresponding values of i for which n is k-hyperperfect:

The newly introduced mathematical concept of hyperdeficiency is related to the hyperperfect numbers.

Definition (Minoli 2010): For any integer n and for integer k, ${\displaystyle k>0}$, define the k-hyperdeficiency (or simply the hyperdeficiency) for the number n as

   δk(n) = n(k+1) +(k-1) – kσ(n)

A number n is said to be k-hyperdeficient if δ_k(n) > 0.

Note that for k=1 one gets δ₁(n)= 2n–σ(n), which is the standard traditional definition of deficiency.

Lemma: A number n is k-hyperperfect (including k=1) if and only if the k-hyperdeficiency of n, δ_k(n) = 0.

Lemma: A number n is k-hyperperfect (including k=1) if and only if for some k, δ_k-j(n) = -δ_k+j(n) for at least one j > 0.

• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. p. 114. ISBN 1-4020-4215-9. Zbl 1151.11300.

• MathWorld: Hyperperfect number
• A long list of hyperperfect numbers under Data