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: ${\ displaystyle \ sigma \,}$

n is a super-perfect number if and only if

{\ displaystyle \ sigma (\ sigma (n)) = 2n. \,}

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). ${\ displaystyle \ sigma (n) = 2n. \,}$ ${\ displaystyle n \ mapsto \ sigma (n) -n}$

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 ${\ displaystyle n}$	${\ displaystyle \ sigma (n)}$	${\ displaystyle \ sigma (\ sigma (n))}$	${\ displaystyle 2n}$	Super perfect?
${\ displaystyle 2}$	${\ displaystyle \ sigma (2) = 1 + 2 = 3}$	${\ displaystyle \ sigma (3) = 1 + 3 = 4}$	${\ displaystyle 4}$	Yes
${\ displaystyle 3}$	${\ displaystyle \ sigma (3) = 1 + 3 = 4}$	${\ displaystyle \ sigma (4) = 1 + 2 + 4 = 7}$	${\ displaystyle 6}$	No
${\ displaystyle 6}$	${\ displaystyle \ sigma (6) = 1 + 2 + 3 + 6 = 12}$	${\ displaystyle \ sigma (12) = 1 + 2 + 3 + 4 + 6 + 12 = 28}$	${\ displaystyle 12}$	No
${\ displaystyle 8}$	${\ displaystyle \ sigma (8) = 1 + 2 + 4 + 8 = 15}$	${\ displaystyle \ sigma (15) = 1 + 3 + 5 + 15 = 24}$	${\ displaystyle 16}$	No
${\ displaystyle 10}$	${\ displaystyle \ sigma (10) = 1 + 2 + 5 + 10 = 18}$	${\ displaystyle \ sigma (18) = 1 + 2 + 3 + 6 + 9 + 18 = 39}$	${\ displaystyle 20}$	No
${\ displaystyle 16}$	${\ displaystyle \ sigma (16) = 1 + 2 + 4 + 8 + 16 = 31}$	${\ displaystyle \ sigma (31) = 1 + 31 = 32}$	${\ displaystyle 32}$	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. ${\ displaystyle 2 ^ {p-1}}$ ${\ displaystyle 2 ^ {p} -1}$

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.

{\ displaystyle \ sigma ^ {m} (n) = k \ cdot n}

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: ${\ displaystyle (m, k)}$

The number 21 is a -super-perfect number because: ${\ displaystyle (2,3)}$

{\ displaystyle \ sigma (21) = 1 + 3 + 7 + 21 = 32}

{\ displaystyle \ sigma ^ {m} (21) = \ sigma ^ {2} (21) = \ sigma (\ sigma (21)) = \ sigma (32) = 1 + 2 + 4 + 8 + 16 + 32 = 63}

But it is also .

{\ displaystyle k \ times 21 = 3 \ times 21 = 63}

The number 14 is a super perfect number because: ${\ displaystyle (3.12)}$

{\ displaystyle \ sigma (14) = 1 + 2 + 7 + 14 = 24}

{\ displaystyle \ sigma ^ {2} (14) = \ sigma (\ sigma (14)) = \ sigma (24) = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60}

{\ displaystyle \ sigma ^ {m} (14) = \ sigma ^ {3} (14) = \ sigma (\ sigma (\ sigma (14))) = \ sigma (60) = 1 + 2 + 3 + 4 + 5 + 6 + 10 + 12 + 15 + 20 + 30 + 60 = 168}

But it is also .

{\ displaystyle k \ times 14 = 12 \ times 14 = 168}

The number 18 is a super perfect number because: ${\ displaystyle (4.20)}$

{\ displaystyle \ sigma (18) = 1 + 2 + 3 + 6 + 9 + 18 = 39}

{\ displaystyle \ sigma ^ {2} (18) = \ sigma (\ sigma (18)) = \ sigma (39) = 1 + 3 + 13 + 39 = 56}

{\ displaystyle \ sigma ^ {3} (18) = \ sigma (\ sigma (\ sigma (18))) = \ sigma (56) = 1 + 2 + 4 + 7 + 8 + 14 + 28 + 56 = 120 }

{\ displaystyle \ sigma ^ {m} (18) = \ sigma ^ {4} (18) = \ sigma (\ sigma (\ sigma (\ sigma (\ sigma (18)))) = \ sigma (120) = 1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 + 120 = 360}

But it is also .

{\ displaystyle k \ times 18 = 20 \ times 18 = 360}

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)