A number such that
where is the divisor function is called a superperfect number. Even superperfect numbers are just , where is a Mersenne prime. If any odd superperfect numbers exist, they are square numbers and either or is divisible by at least three distinct primes.
More generally, an -superperfect (or (, 2)-superperfect) number is a number for which , and an -perfect number is a number for which . A number can be tested to see if it is -perfect using the following Wolfram Language code:
SuperperfectQ[m_, n_, k_:2] := Nest[DivisorSigma[1, #]&, n, m] == k n
The first few (2, 2)-perfect numbers are 2, 4, 16, 64, 4096, 65536, 262144, ... (OEIS A019279; Cohen and te Riele 1996). For , there are no even -superperfect numbers (Guy 1994, p. 65). On the basis of computer searches, J. McCranie has shown that there are no -superperfect numbers less than for any (McCranie, pers. comm., Nov. 11, 2001). McCranie further believes that there are no -superperfect numbers for , since for all in that range