Superperfect number

In number theory, a superperfect number is a positive integer $n$ that satisfies


 * $$\sigma^2(n)=\sigma(\sigma(n))=2n\, ,$$

where $σ$ is the divisor summatory 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 σ(16) = 1 + 2 + 4 + 8 + 16 = 31, and σ(31) = 1 + 31 = 32, thus σ(σ(16)) = 32 = 2 × 16.

If $n$ is an even superperfect number, then $n$ must be a power of 2, $2^{k}$, such that $2^{k+1} − 1$ is a Mersenne prime.

It is not known whether there are any odd superperfect numbers. An odd superperfect number $n$ would have to be a square number such that either $n$ or $σ(n)$ is divisible by at least three distinct primes. There are no odd superperfect numbers below 7.

Generalizations
Perfect and superperfect numbers are examples of the wider class of m-superperfect numbers, which satisfy


 * $$ \sigma^m(n) = 2n, $$

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


 * $$\sigma^m(n)=kn\, .$$

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. Examples of classes of (m,k)-perfect numbers are:


 * {| class="wikitable"

! m ! k ! (m,k)-perfect numbers ! OEIS sequence
 * 2
 * 2
 * 2, 4, 16, 64, 4096, 65536, 262144
 * 2
 * 3
 * 8, 21, 512
 * 2
 * 4
 * 15, 1023, 29127
 * 2
 * 6
 * 42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024
 * 2
 * 7
 * 24, 1536, 47360, 343976
 * 2
 * 8
 * 60, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072
 * 2
 * 9
 * 168, 10752, 331520, 691200, 1556480, 1612800, 106151936
 * 2
 * 10
 * 480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296
 * 2
 * 11
 * 4404480, 57669920, 238608384
 * 2
 * 12
 * 2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120
 * 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, ...
 * }
 * 10
 * 480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296
 * 2
 * 11
 * 4404480, 57669920, 238608384
 * 2
 * 12
 * 2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120
 * 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, ...
 * }
 * 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, ...
 * }
 * 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 26, 32, 39, 42, 60, 65, 72, 84, 96, 160, 182, ...
 * }
 * }