A superperfect number satisfies \(\sigma(\sigma(n)) = 2n\), where \(\sigma\) is the sum-of-divisors function — applying the perfect-number condition one level deeper. The sequence is 2, 4, 16, 64, 4096, 65536, 262144, 1073741824.
The concept, introduced by D. Suryanarayana in 1969, ties straight back to [[mersenne-prime]] theory: the even superperfect numbers are exactly \(2^{p-1}\) where \(2^p - 1\) is a Mersenne prime — so they're in one-to-one correspondence with the even [[perfect-number]]s. Whether any odd superperfect number exists is, like the odd perfect number question, unresolved (none is known, and any would have to be a perfect square).