Superperfect number

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 sum-of-divisors 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, 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 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, 1073741824, 1152921504606846976	A019279
2	3	8, 21, 512	A019281
2	4	15, 1023, 29127, 355744082763	A019282
2	6	42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024, 22548578304	A019283
2	7	24, 1536, 47360, 343976, 572941926400	A019284
2	8	60, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072, 7635468288, 16106127360, 711488165526, 1098437885952, 1422976331052	A019285
2	9	168, 10752, 331520, 691200, 1556480, 1612800, 106151936, 5099962368, 4010593484800	A019286
2	10	480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296, 14763499520, 38385098752	A019287
2	11	4404480, 57669920, 238608384	A019288
2	12	2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120, 16785793024, 22648550400, 36051025920, 51001180160, 144204103680	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.
• Cohen, G. L.; te Riele, H. J. J. (1996). "Iterating the sum-of-divisors function". Experimental Mathematics . 5 (2): 93–100. doi:10.1080/10586458.1996.10504580. S2CID   28197771. 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.