Superperfect number

Short description: Number whose divisors summed twice over equal twice itself

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

$\displaystyle{ \sigma^2(n)=\sigma(\sigma(n))=2n\, , }$

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).[1]

The first few superperfect numbers are :

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... (sequence A019279 in the OEIS).

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, 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 7×1024.[1]

Generalizations

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

$\displaystyle{ \sigma^m(n) = 2n , }$

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]

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

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, 262144 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. Guy (2004) p. 99.
2. Cohen & te Riele (1996)
3. Guy (2007) p.79