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In mathematics, a '''superperfect number''' is a positive [[integer]] ''n'' that satisfies

:<math>\sigma^2(n)=\sigma(\sigma(n))=2n\, ,</math>

where σ is the [[divisor summatory function]]. Superperfect numbers are a generalization of [[perfect number]]s.  The term was coined by D. Suryanarayana (1969).<ref name=Guy99/>

The first few superperfect numbers are :

:[[2 (number)|2]], [[4 (number)|4]], [[16 (number)|16]], [[64 (number)|64]], [[4096 (number)|4096]], [[65536 (number)|65536]], 262144, 1073741824, ... {{OEIS|id=A019279}}.

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<sup>''k''</sup>, such that 2<sup>''k''+1</sup> − 1 is a [[Mersenne prime]].<ref name="Guy99" /><ref name="mathworld">{{MathWorld|urlname=SuperperfectNumber |title=Superperfect Number }}</ref>

It is not known whether there are any [[Odd number|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. <ref name=mathworld/> There are no odd superperfect numbers below 7{{e|24}}.<ref name=Guy99>Guy (2004) p. 99.</ref>

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

:<math> \sigma^m(n) = 2n , </math>

corresponding to ''m''=1 and 2 respectively.  For ''m'' ≥ 3 there are no even ''m''-superperfect numbers.<ref name=Guy99/>

The ''m''-superperfect numbers are in turn examples of (''m'',''k'')-perfect numbers which satisfy<ref>Cohen & te Riele (1996)</ref>

:<math>\sigma^m(n)=kn\, .</math>

With this notation, perfect numbers are (1,2)-perfect, [[multiperfect number]]s are (1,''k'')-perfect,  superperfect numbers are (2,2)-perfect and ''m''-superperfect numbers are (''m'',2)-perfect.<ref>Guy (2007) p.79</ref>  Examples of classes of (''m'',''k'')-perfect numbers are:

:{| class="wikitable"
|-
! ''m''
! ''k''
! (''m'',''k'')-perfect numbers
! [[OEIS]] sequence
|-
| 2
| 2
| 2, 4, 16, 64, 4096, 65536, 262144
| {{OEIS link|A019279}}
|-
| 2
| 3
| 8, 21, 512
| {{OEIS link|A019281}}
|-
| 2
| 4
| 15, 1023, 29127
| {{OEIS link|A019282}}
|-
| 2
| 6
| 42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024 
| {{OEIS link|A019283}}
|-
| 2
| 7
| 24, 1536, 47360, 343976
| {{OEIS link|A019284}}
|-
| 2
| 8
| 60, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072 
| {{OEIS link|A019285}}
|-
| 2
| 9
| 168, 10752, 331520, 691200, 1556480, 1612800, 106151936 
| {{OEIS link|A019286}}
|-
| 2
| 10
| 480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296
| {{OEIS link|A019287}}
|-
| 2
| 11
| 4404480, 57669920, 238608384 
| {{OEIS link|A019288}}
|-
| 2
| 12
| 2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120  
| {{OEIS link|A019289}}
|-
| 3
| any
| 12, 14, 24, 52, 98, 156, 294, 684, 910, 1368, 1440, 4480, 4788, 5460, 5840, ...  
| {{OEIS link|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, ...  
| {{OEIS link|A019293}}
|}

== Notes ==
{{reflist}}

== References ==
{{refbegin}}
*{{PlanetMath |urlname=SuperperfectNumber |title=Superperfect Number}}
*{{cite journal |first1=G. L. |last1=Cohen |first2=H. J. J. |last2=te Riele |title=Iterating the sum-of-divisors function |journal=[[Experimental Mathematics (journal)|Experimental Mathematics]] |volume=5 |year=1996 |pages=93–100 |zbl=0866.11003 |doi=10.1080/10586458.1996.10504580}}
* {{cite book |last=Guy |first=Richard K. |authorlink=Richard K. Guy |title=Unsolved problems in number theory |publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 |zbl=1058.11001 |at=B9}}
* {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 }}
* {{cite journal |zbl=0165.36001 |last=Suryanarayana |first=D. |title=Super perfect numbers |journal=Elem. Math. |volume=24 |pages=16–17 |year=1969}}
{{refend}}


{{Divisor classes}}
{{Classes of natural numbers}}

[[Category:Divisor function]]
[[Category:Integer sequences]]
[[Category:Unsolved problems in mathematics]]