A -superperfect number is an integer such that , where is the iterated sum of divisors function. For example, 16 is 2-superperfect since its divisors add up to 31, and in turn the divisors of 31 add up to 32, which is twice 16.
At first Suryanarayana only considered 2-superperfect numbers. It is easy to prove that numbers of the form are 2-superperfect only if is a Mersenne prime. The existence of odd 2-superperfect numbers appears as unlikely as that of regular odd perfect numbers.
Later, Dieter Bode generalized the concept for any and proved that when there are no even -superperfect numbers. Others have further generalized the concept to -superperfect numbers satisifying the equality , and Weisstein programs a Mathematica command to default to when the third argument is omitted. For example, 8, 21, and 512 are (2, 3)-superperfect, since the second iteration of the sum of divisors function gives thrice them, 24, 63, and 1536 respectively.
Not to be confused with hyperperfect numbers, which satisfy a different equality involving the sum of divisors function.