# Superperfect number

In mathematics, a superperfect number is a positive integer n that satisfies

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

where σ is the divisor function. Superperfect numbers are a generalization of perfect numbers. The term was coined by Suryanarayana (1969).[1]

The first few superperfect numbers are

2, 4, 16, 64, 4096, 65536, 262144 (sequence A019279 in OEIS).

If n is an even superperfect number then n must be a power of 2, 2k, such that 2k+1-1 is a Mersenne prime.[1][2]

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.[2] There are no odd superperfect numbers below 7x1024.[1]

## Generalisations

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.[1]

The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfy[3]

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

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.[4] 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, 26144 A019279
2 3 8, 21, 512 A019281
2 4 15, 1023, 29127 A019282
2 6 42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024 A019283
2 7 24, 1536, 47360, 343976 A019284
2 8 60, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072 A019285
2 9 168, 10752, 331520, 691200, 1556480, 1612800, 106151936 A019286
2 10 480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296 A019287
2 11 4404480, 57669920, 238608384 A019288
2 12 2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120 A019289
3 any 12, 14, 24, 52, 98, 156, 294, 684, 910, 1368, 1440, 4480, 4788, 5460, 5840, ... A019292
4 any 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 26, 32, 39, 42, 60, 65, 72, 84, 96, 160, 182, ... A019293

## Notes

1. ^ a b c d Guy (2004) p.99
2. ^ a b
3. ^ Cohen & te Riele (1996)
4. ^ Guy (2007) p.79