Superperfect number explained

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

\sigma2(n)=\sigma(\sigma(n))=2n,

where 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, 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]

Generalizations

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:

mk(m,k)-perfect numbersOEIS sequence
222, 4, 16, 64, 4096, 65536, 262144
238, 21, 512
2415, 1023, 29127
2642, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024
2724, 1536, 47360, 343976
2860, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072
29168, 10752, 331520, 691200, 1556480, 1612800, 106151936
210480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296
2114404480, 57669920, 238608384
2122200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120
3any12, 14, 24, 52, 98, 156, 294, 684, 910, 1368, 1440, 4480, 4788, 5460, 5840, ...
4any2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 26, 32, 39, 42, 60, 65, 72, 84, 96, 160, 182, ...

References

Notes and References

  1. Guy (2004) p. 99.
  2. Cohen & te Riele (1996)
  3. Guy (2007) p.79