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Superperfect number

{{short description|Number whose divisors summed twice over equal twice itself}}

In number theory, a superperfect number is a positive integer {{mvar|n}} that satisfies

:\sigma^2(n)=\sigma(\sigma(n))=2n\, ,

where {{mvar|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).

The first few superperfect numbers are:

:2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... {{OEIS|id=A019279}}.

To illustrate: it can be seen that 16 is a superperfect number as {{nowrap|1=σ(16) = 1 + 2 + 4 + 8 + 16 = 31}}, and {{nowrap|1=σ(31) = 1 + 31 = 32}}, thus {{nowrap|1=σ(σ(16)) = 32 = 2 × 16}}.

If {{mvar|n}} is an even superperfect number, then {{mvar|n}} must be a power of 2, {{math|2k}}, such that {{math|2k+1 − 1}} is a Mersenne prime.{{MathWorld|urlname=SuperperfectNumber |title=Superperfect Number }}

It is not known whether there are any odd superperfect numbers. An odd superperfect number {{mvar|n}} would have to be a square number such that either {{mvar|n}} or {{math|σ(n)}} is divisible by at least three distinct primes. There are no odd superperfect numbers below 7{{e|24}}.Guy (2004) p. 99.

Generalizations

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.

The m-superperfect numbers are in turn examples of (m,k)-perfect numbers which satisfyCohen & te Riele (1996)

:\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.Guy (2007) p.79 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, 1073741824, 1152921504606846976

| {{OEIS link|A019279}}

2

| 3

| 8, 21, 512

| {{OEIS link|A019281}}

2

| 4

| 15, 1023, 29127, 355744082763

| {{OEIS link|A019282}}

2

| 6

| 42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024, 22548578304

| {{OEIS link|A019283}}

2

| 7

| 24, 1536, 47360, 343976, 572941926400

| {{OEIS link|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

| {{OEIS link|A019285}}

2

| 9

| 168, 10752, 331520, 691200, 1556480, 1612800, 106151936, 5099962368, 4010593484800

| {{OEIS link|A019286}}

2

| 10

| 480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296, 14763499520, 38385098752

| {{OEIS link|A019287}}

2

| 11

| 4404480, 57669920, 238608384

| {{OEIS link|A019288}}

2

| 12

| 2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120, 16785793024, 22648550400, 36051025920, 51001180160, 144204103680

| {{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 | volume=5 |issue=2 |year=1996 |pages=93–100 |zbl=0866.11003 |doi=10.1080/10586458.1996.10504580|s2cid=28197771 |url=https://ir.cwi.nl/pub/10355 }}
  • {{cite book |last=Guy |first=Richard K. |author-link=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 number theory