Superperfect number

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Divisibility-based
sets of integers
Form of factorization:
Prime number
Composite number
Powerful number
Square-free number
Achilles number
Constrained divisor sums:
Perfect number
Almost perfect number
Quasiperfect number
Multiply perfect number
Hyperperfect number
Superperfect number
Unitary perfect number
Semiperfect number
Primitive semiperfect number
Practical number
Numbers with many divisors:
Abundant number
Highly abundant number
Superabundant number
Colossally abundant number
Highly composite number
Superior highly composite number
Other:
Deficient number
Weird number
Amicable number
Friendly number
Sociable number
Solitary number
Sublime number
Harmonic divisor number
Frugal number
Equidigital number
Extravagant number
See also:
Divisor function
Divisor
Prime factor
Factorization
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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 generalisation of perfect numbers.

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]

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

Perfect and superperfect numbers are examples of the wider class of (m,k)-perfect numbers which satisfy

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

With this notation, perfect numbers are (1,2)-perfect and superperfect numbers are (2,2)-perfect. Other classes of (m,k)-perfect numbers are:

m k (m,k)-perfect numbers OEIS sequence
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

[edit] References

  1. ^ a b Eric W. Weisstein, Superperfect Number at MathWorld.
  2. ^ Problem B9 in Richard K. Guy's Unsolved Problems in Number Theory (ISBN 0-387-94289-0)