Super perfect number
A natural number n is called a super-perfect number if the sum of the divisors of the sum of its divisors is twice the original number n . If you use the notation for the partial sum function , you can write down the definition as follows:
- n is a super-perfect number if and only if
The more well-known perfect numbers , on the other hand, satisfy the equation. The question of whether a number is super perfect arises when examining the iterated partial sum function (see also content chain ; here, however, the figure is iterated).
Examples and characteristics
The number 6 has the divisors 1, 2, 3 and 6. The sum of these numbers is 12. The divisors of 12 in turn are 1, 2, 3, 4, 6 and 12, the sum of which is 28. Because 28 ≠ 2 · 6, 6 is not a super-perfect number. Further calculation examples are:
number | Super perfect? | |||
---|---|---|---|---|
Yes | ||||
No | ||||
No | ||||
No | ||||
No | ||||
Yes |
The first super-perfect numbers are 2, 4, 16, 64, 4096, 65536, 262144, ... (sequence A019279 in OEIS ).
Every even super-perfect number has the form , where is a Mersenne prime (example: 16 is super perfect and 31 is a Mersenne prime). Conversely, every Mersenne prime yields an even, super-perfect number. It is not known whether there are odd super-perfect numbers.
generalization
Like perfect numbers, super- perfect numbers are examples of numbers in the upper class of ( m , k ) -super-perfect numbers, which are defined as follows:
- n is an ( m , k ) -super-perfect number if and only if holds.
So perfect numbers are (1,2) -super-perfect and super-perfect numbers are (2,2) -super-perfect. The mathematicians GL Cohen and HJJ te Riele consider it possible that every number is ( m , k ) -super perfect for suitable m and k .
Here are a few examples of generalized -super-perfect numbers:
The number 21 is a -super-perfect number because:
- But it is also .
The number 14 is a super perfect number because:
- But it is also .
The number 18 is a super perfect number because:
- But it is also .
Further examples of ( m , k ) -super-perfect numbers follow :
m | k | ( m , k ) -super-perfect numbers | OEIS episode |
---|---|---|---|
2 | 2 | 2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864 | Follow A019279 in OEIS |
2 | 3 | 8, 21, 512 | Follow A019281 in OEIS |
2 | 4th | 15, 1023, 29127, 355744082763 | Follow A019282 in OEIS |
2 | 6th | 42, 84, 160, 336, 1344, 86016, 550095, 1376256, 5505024, 22548578304 | Follow A019283 in OEIS |
2 | 7th | 24, 1536, 47360, 343976 | Follow A019284 in OEIS |
2 | 8th | 60, 240, 960, 4092, 16368, 58254, 61440, 65472, 116508, 466032, 710400, 983040, 1864128, 3932160, 4190208, 67043328, 119304192, 268173312, 1908867072, 7635468288, 16106127360 | Follow A019285 in OEIS |
2 | 9 | 168, 10752, 331520, 691200, 1556480, 1612800, 106151936, 5099962368 | Follow A019286 in OEIS |
2 | 10 | 480, 504, 13824, 32256, 32736, 1980342, 1396617984, 3258775296, 14763499520, 38385098752 | Follow A019287 in OEIS |
2 | 11 | 4404480, 57669920, 238608384 | Follow A019288 in OEIS |
2 | 12 | 2200380, 8801520, 14913024, 35206080, 140896000, 459818240, 775898880, 2253189120, 16785793024, 22648550400, 36051025920, 51001180160, 144204103680 | Follow A019289 in OEIS |
3 | k | 1, 12, 14, 24, 52, 98, 156, 294, 684, 910, 1368, 1440, 4480, 4788, 5460, 5840, 6882, 7616, 9114, 14592, 18288, 22848, 32704, 40880, 52416, 53760, 54864, 56448, 60960, 65472, 94860, 120960, 122640, 169164, 185535, 186368, 194432 | Follow A019292 in OEIS |
4th | k | 1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 26, 32, 39, 42, 60, 65, 72, 84, 96, 160, 182, 336, 455, 512, 896, 960, 992, 1023, 1280, 1536, 1848, 2040, 2688, 4092, 5920, 7808, 7936, 10416, 16352, 20384, 21824, 23424, 24564, 29127, 33792, 41440 | Follow A019293 in OEIS |
literature
- D. Suryanarayana: Super perfect numbers . In: Elements of Mathematics , 24, 16–17, 1969, digizeitschriften.de
- Dieter Bode: About a generalization of perfect numbers . Dissertation, Braunschweig 1971
- Richard K. Guy: Unsolved Problems in Number Theory . 3. Edition. Springer, 2004, Chapters B2 and B9, Google books
- GL Cohen, HJJ te Riele: Iterating the sum-of-divisors function . In: Experimental Mathematics , 5, 93-100, 1993, projecteuclid.org
Web links
- Eric W. Weisstein : Superperfect Number . In: MathWorld (English).
- Superperfect Number . In: PlanetMath . (English)