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 summatory 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, 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 x 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.^[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, 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. Weisstein, Eric W. "Superperfect Number". MathWorld.
3. Cohen & te Riele (1996)
4. Guy (2007) p.79

References

• Superperfect Number at PlanetMath.org.
• Cohen, G. L.; te Riele, H. J. J. (1996). "Iterating the sum-of-divisors function". Experimental Mathematics. 5: 93–100. doi:10.1080/10586458.1996.10504580. Zbl 0866.11003.
• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B9. ISBN 978-0-387-20860-2. Zbl 1058.11001.
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.
• Suryanarayana, D. (1969). "Super perfect numbers". Elem. Math. 24: 16–17. Zbl 0165.36001.