Superperfect Number

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

REFERENCES:

Cohen, G. L. and te Riele, J. J. "Iterating the Sum-of-Divisors Function." Experim. Math. 5, 93-100, 1996.

Guy, R. K. "Superperfect Numbers." §B9 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 65-66, 1994.

Kanold, H.-J. "Über 'Super Perfect Numbers.' " Elem. Math. 24, 61-62, 1969.

Lord, G. "Even Perfect and Superperfect Numbers." Elem. Math. 30, 87-88, 1975.

Sloane, N. J. A. Sequence A019279 in "The On-Line Encyclopedia of Integer Sequences."

Suryanarayana, D. "Super Perfect Numbers." Elem. Math. 24, 16-17, 1969.

Suryanarayana, D. "There Is No Odd Super Perfect Number of the Form ." Elem. Math. 24, 148-150, 1973.