In number theory, a superperfect number is a positive integer that satisfies
\sigma2(n)=\sigma(\sigma(n))=2n,
where is the sum-of-divisors function. Superperfect numbers are not a generalization of perfect numbers but have a common generalization. The term was coined by D. Suryanarayana (1969).
The first few superperfect numbers are :
2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... .
To illustrate: it can be seen that 16 is a superperfect number as, and, thus .
If is an even superperfect number, then must be a power of 2,, such that is a Mersenne prime.
It is not known whether there are any odd superperfect numbers. An odd superperfect number would have to be a square number such that either or is divisible by at least three distinct primes. There are no odd superperfect numbers below 7.[1]
Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy
\sigmam(n)=2n,
corresponding to m = 1 and 2 respectively. For m ≥ 3 there are no even m-superperfect numbers.
The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy[2]
\sigmam(n)=kn.
With this notation, perfect numbers are (1,2)-perfect, multiperfect numbers are (1,k)-perfect, superperfect numbers are (2,2)-perfect and m-superperfect numbers are (m,2)-perfect.[3] Examples of classes of (m,k)-perfect numbers are:
| m | k | (m,k)-perfect numbers | OEIS sequence | |
|---|---|---|---|---|
| 2 | 2 | 2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976 | ||
| 2 | 3 | 8, 21, 512 | ||
| 2 | 4 | 15, 1023, 29127, 355744082763 | ||
| 2 | 6 | 42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024, 22548578304 | ||
| 2 | 7 | 24, 1536, 47360, 343976, 572941926400 | ||
| 2 | 8 | 60, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072, 7635468288, 16106127360, 711488165526, 1098437885952, 1422976331052 | ||
| 2 | 9 | 168, 10752, 331520, 691200, 1556480, 1612800, 106151936, 5099962368, 4010593484800 | ||
| 2 | 10 | 480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296, 14763499520, 38385098752 | ||
| 2 | 11 | 4404480, 57669920, 238608384 | ||
| 2 | 12 | 2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120, 16785793024, 22648550400, 36051025920, 51001180160, 144204103680 | ||
| 3 | any | 12, 14, 24, 52, 98, 156, 294, 684, 910, 1368, 1440, 4480, 4788, 5460, 5840, ... | ||
| 4 | any | 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 26, 32, 39, 42, 60, 65, 72, 84, 96, 160, 182, ... |