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.
