Content chain

Under an aliquot sequence (also aliquot sequence of English. Aliquot sequence ) is defined as a sequence of positive integers in which each of the number of content (the sum of the proper divisors ) of its predecessor's.

Formal definition

The aliquot sequence with the start value n or aliquot sequence of n is the result

${\displaystyle {\bigl (}n,\,s(n),\,s(s(n)),\,s(s(s(n))),\,\ldots {\bigr )},}$

where with the partial sum . ${\displaystyle s(n)=\sigma (n)-n}$ ${\displaystyle \sigma }$

properties

Natural numbers on the content strings in the same prime number (except for the 0 and 1) lead, form a prime family (Engl. Prime family ), also briefly P family (Engl. P-family ) called. A ring family (English cycle family ), also called R-family (English c-family ), terminates in a ring of perfect , friendly or sociable numbers .

Perfect numbers terminate in a perfect number, namely themselves (because they are so defined).

Friendly numbers terminate in a cycle with a length of 2 (because they are so defined).

Sociable numbers terminate in a cycle of length 3 or greater (because they are so defined).

Content chains can for example be generated in the factoring database .

The Catalan-Dickson conjecture (named after Eugène Charles Catalan and Leonard Eugene Dickson ) states that every chain of contents becomes periodic or ends with 0. To this day it has neither been proven nor refuted. The mathematicians Richard Kenneth Guy and John L. Selfridge assume, however, that the Catalan-Dickson conjecture is false (which would mean that there are numbers whose content strings do not end in a 0, a perfect number, or a cycle ; their content chain would thus be infinitely long).

A number that in no aliquot sequence occurs (except as a starting value of the content of its own chain) is called untouchable number (from the English untouchable number ).

Examples

Example 1:

The content chain of 10 is (10, 8, 7, 1, 0), thus has a length of n = 5 and terminates in the 0:

s (10) = 1 + 2 + 5 = 8

s ( 08) = 1 + 2 + 4 = 7

s ( 07) = 1

s ( 01) = 0

Example 2:

The content chain of 95 is (95, 25, 6, 6, ...), thus has a length of n = 3 and terminates in the perfect number 6:

s (95) = 1 + 5 + 19 = 25

s (25) = 1 + 5 = 6

s ( 06) = 1 + 2 + 3 = 6

s ( 06) = 1 + 2 + 3 = 6

...

Example 3:

The content chain of 220 is ( 220 , 284, 220 , 284, 220 , ...), has a length of n = 2 and terminates in a cycle with a length of 2 (220 and 284 are friendly numbers ):

s ( 220 ) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

s (284) = 1 + 2 + 4 + 71 + 142 = 220

s ( 220 ) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

s (284) = 1 + 2 + 4 + 71 + 142 = 220

...

Well over 363000 friendly pairs of numbers are known.

Example 4:

The content chain of 12496 is ( 12496 , 14288, 15472, 14536, 14264, 12496 , ...), has a length of n = 5 and terminates in a cycle with a length of 5 (these 5 numbers are social numbers ):

s ( 12496 ) = 1 + 2 + 4 + 8 + 11 + 16 + 22 + 44 + 71 + 88 + 142 + 176 + 284 + 568 + 781 + 1136 + 1562 + 3124 + 6248 = 14288

s (14288) = 1 + 2 + 4 + 8 + 16 + 19 + 38 + 47 + 76 + 94 + 152 + 188 + 304 + 376 + 752 + 893 + 1786 + 3572 + 7144 = 15472

s (15472) = 1 + 2 + 4 + 8 + 16 + 967 + 1934 + 3868 + 7736 = 14536

s (14536) = 1 + 2 + 4 + 8 + 23 + 46 + 79 + 92 + 158 + 184 + 316 + 632 + 1817 + 3634 + 7268 = 14264

s (14264) = 1 + 2 + 4 + 8 + 1783 + 3566 + 7132 = 12496

...

This cycle of length 5 is the only known one. For example, the content chains of 9464, 12032, 12496, 14264, 14288, 14536, 15472, 15476, 16312, 18922, ... terminate in this cycle.

Example 5:

The content chain of 14316 is ( 14316 , 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946 48976, 45946, 22976, 22744, 19916, 17716, 14316 , ...) and terminated in a cycle with a length of 28 (these 28 numbers are thus also sociable numbers ).

This cycle of length 28 is the only known one. For example, the content chains of 2856, 3360, 5784, 5916, 7524, 7860, 8736, 9052, 9204, 10328, 14316, 17496, ... terminate in this cycle.

further examples:

• The content chain lengths for the first 50 numbers n = 1, 2, 3, ... are:

2, 3, 3, 4, 3, 1, 3, 4, 5, 5, 3, 8, 3, 6, 6, 7, 3, 5, 3, 8, 4, 7, 3, 6, 2, 8, 4, 1, 3, 16 , 3, 4, 7, 9, 4, 5, 3, 8, 4, 5, 3, 15, 3, 6, 8, 9, 3, 7, 5, 4, ... (Follow A098007 in OEIS )

Example:

The number 16 is in the 30th position in the above list. This means that the content chain of n = 30 has a length of 16.

• If you do not add the start value to the content chains, the content chain lengths for the first 50 numbers n = 1, 2, 3, ... are as follows:

1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 1, 2, 15 , 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, ... (Follow A044050 in OEIS )

Example:

The number 15 is in the 30th position of the above list. This means that the content chain of n = 30 has a length of 15 if the starting number n = 30 is not included.

You always get a length that is 1 smaller than in the previous list, unless it is a content chain that does not terminate in 0.

• The following list shows the number in which the chain of contents ends before it becomes 1 and then 0 (with the exception of n = 1):

1, 2, 3, 3, 5, 6, 7, 7, 3, 7, 11, 3, 13, 7, 3, 3, 17, 11, 19, 7, 11, 7, 23, 17, 6, 3, 13, 28, 29, 3 , 31, 31, 3, 7, 13, 17, 37, 7, 17, 43, 41, 3, 43, 43, 3, 3, 47, 41, 7, 43, ... (Follow A115350 in OEIS )

Example:

The number 3 is in the 30th position of the above list. This means that the content chain of n = 30 ends in the prime number 3, followed by 1 and 0.

• Now follows the list of numbers n, the content of which ends in 1, followed by 0:

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33 , 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, ... (Follow A080907 in OEIS )

Example:

In the 30th position of the above list is the number n = 33. This means that 30 of the first 33 content chains lead to 1.

Conversely, it means that only 3 of the first 33 content chains do not end in 1, so they have to end in a perfect number or a cycle (in this case the three numbers 6, 25 and 28).

• The following is the list of numbers that terminate in a perfect number (i.e. 6, 28, 496, 8128, ...) but are not themselves perfect:

25, 95, 119, 143, 417, 445, 565, 608, 650, 652 , 675, 685, 783, 790, 909, 913, ... (sequence A063769 in OEIS )

Example:

In the 10th position of the above list is the number n = 652. The chain of contents of this number ends (already at the 2nd position) in the perfect number 496.

• The following is the list of numbers whose content chain ends in a cycle that has a length of at least 2 (one also says " chain of the order (of) at least 2 "):

220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, 2542, 2620, 2630, 2652, 2676, 2678, 2856 , 2924, 2930, 2950, ​​2974, 3124, 3162, 3202, 3278, 3286, 3332, 3350, 3360, ... (follow A121507 in OEIS )

Example:

In the 30th position of the above list is the number n = 2856. The chain of contents of this number ends in the (only known) cycle of length 28, starting with the number 14316.

• Finally, there is a list of numbers whose content chains are not yet fully known because the values ​​in them could not be factored:

276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488 , 1512, 1560, 1572, 1578, 1590, 1632, 1650, 1662, 1674, 1722, 1734, 1758, 1770, 1806, 1836, ... (series A131884 in OEIS )

Example:

In the 30th position of the above list is the number n = 1488. The chain of contents of this number is known and factored up to the 1621st position. The 1622nd position of this chain of contents is a 183-digit number that has not yet been factored.

Lehmer-Six and Lehmer-Five

The first six open (not fully calculated) chains in the interval [1, 1000] were named Lehmer-Six after the couple Derrick Henry Lehmer and Emma Lehmer . Their starting numbers were 276, 552, 564, 660, 840 and 966.

The chain with the starting number 840 is now fully known. It terminates at the 746th position in the prime number 601, followed by 1 and 0. The remaining 5 open chains are now called Lehmer-Five . The current status can be found in the following table (status: September 15, 2019):

Starting number ${\displaystyle n}$	calculated up to index ${\displaystyle i=}$	Number of digits	Content number that has not yet been fully factored ${\displaystyle s^{i}(n)}$	link
			previously known factorization, residual factor has ... digits ${\displaystyle Z}$
276	2140	213	63507972071398516677404354289896457938723327730993453569365104709178716789761584803425032204439982173315795449006502742320455529633799262115212744914825560983612091662291482558098361209166229148255609836120916622
			${\displaystyle s^{2140}(276)=2\cdot 3\cdot Z_{213}}$with a 213-digit composite remaining factor ${\displaystyle Z_{213}}$
552	1142	194	275942402638802963018158551164231236943108760720600623205251990309022691514500286760430711200683243265534593031028085031396636556205345272104952652355662808592899559905512672120
			${\displaystyle s^{1142}(552)=2^{2}\cdot 3^{2}\cdot 5^{2}\cdot 341597\cdot Z_{185}}$ with a 185-digit composite residual factor ${\displaystyle Z_{185}}$
564	3486	198	53889869030924163882566278666753424722399797456860855230736661926105008971872497380069437255224957896372322586809600224095163014784981350921478647920705102983369237847879286
			${\displaystyle s^{3486}(564)=2^{3}\cdot Z_{197}}$ with a 197-digit composite residual factor ${\displaystyle Z_{197}}$
660	1008	200	1854322850496718278726292827819862728794102712408354039969186760580663879074313524447613111805518800152346032546918508762599311217519489496701803401359356743304796844312373206681674330479684431237320
			${\displaystyle s^{1008}(660)=2^{4}\cdot 64373\cdot 232411\cdot 4372211\cdot Z_{182}}$ with a 182-digit composite remaining factor ${\displaystyle Z_{182}}$
966	1035	201	495632203505909769323917659478470131607203616704992223175667065264576553135246665899916282358640765511653578540462117747856794631087721401120803740769564902549910832693304436956490254991080326933044
			${\displaystyle s^{1035}(966)=2^{4}\cdot 11\cdot 47\cdot 53\cdot 5333\cdot Z_{192}}$ with a 192-digit composite remaining factor ${\displaystyle Z_{192}}$

There are another 7 open chains in the interval [1, 1000], namely 306, 396, 696, 780, 828, 888 and 996. However, their chains eventually lead to one of the Lehmer five (i.e. in one of the chains of 276, 552, 564, 660 or 966), namely:

${\displaystyle s^{20}(276)=s^{20}(306)=s^{19}(396)=s^{18}(696)=54684}$

${\displaystyle s^{9}(552)=s^{8}(888)=30984}$

${\displaystyle s^{8}(564)=s^{7}(780)=26376}$

${\displaystyle s^{29}(660)=s^{29}(828)=s^{29}(996)=149838}$

In this respect, these 7 chains do not play a special role and therefore do not belong to the Lehmer Five. The same applies to many chains of higher numbers that are not counted in the following (such as the content chain from 1806, which ends at the 18th position in the chain from 1134).

In the interval [1, 10,000] there are currently 81 open (and, as mentioned above, completely independent) chains, in the interval [1, 100,000] exactly 893 and in the interval [1, 1,000,000] exactly 9127 open chains. In the interval [1, 3,000,000] exactly 27728 open chains (as of December 28, 2019). No name has become established for these chains. The 14 content chains between 1000 and 2000 still open in 1980 were named Godwin fourteen . In the meantime, only 12 content chains are still open in this interval (1248 and 1848 have been completely calculated since then, the content chains of 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920 and 1992 could not yet be completely calculated become).

gallery

Most chains of content end in a prime number. The set of all natural numbers whose content chains end in the same prime number form a prime number family, or family for short . The following gallery shows a few such families. Of course, not all numbers that end in the same prime number can be mapped onto it. In the first picture all numbers ≤ 10000 are shown, whose content chains end in the prime number 3. In the second picture you can only see all the numbers ≤ 1000 whose content strings end in the prime number 3. Families 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 and 43 follow.

• 

  Family 3 to 10000

• 

  Family 3 to 1000

• 

  Family 7 to 1000

• 

  Family 11 to 1000

• 

  Family 13 to 1000

• 

  Family 17 to 1000

• 

  Family 19 to 1000

• 

  Family 23 to 1000

• 

  Family 29 to 1000

• 

  Family 31 to 1000

• 

  Family 37 to 1000

• 

  Family 41 to 1000

• 

  Family 43 to 1000

Here are a few graphs that show how much content chains can grow. Most content chains terminate, i.e. end in 0, in a perfect number or in a cycle. But there are also content chains that have not yet been fully calculated because numbers with over 200 digits appear whose prime factors have not (yet) been calculated (or could not yet be calculated because of their size). These content chains are called open -end chains ( OE chains - nobody knows whether these chains will grow infinitely or whether they will terminate at some point in the 0, in a perfect number or in a cycle). On the x-axis you can see up to where these chains have already been calculated (the values ​​in between are called the content number , starting with 0), the y-axis shows how many positions this content number has.

In the first picture you can see the five OE content chains of the Lehmer Five (276, 552, 564, 660 and 966) together in one graphic. In Fig. 2 to 6 you can see the five OE content chains of the Lehmer Five individually.

This is followed by the OE content chain from 1578, which reaches the previous record low of 56440 at the 1868th position (i.e. only a 5-digit number) and then becomes very large again (at the 7615th position you get a 166-digit number Number that cannot yet be fully factored).

This is followed by the OE content chain of 2340, which can have the highest number of digits in OE content chains to date (at the 790th position you get a 216-digit number that cannot yet be factored completely).

This is followed by the longest known OE content chain, namely from 314718, from which the chain has already been calculated up to the 19026th position and has still not reached an end. At this point you have to factor a 213-digit number, but you haven't made it yet. However, this chain ends at the 6460th position in the 5-digit number 16100, which of course has its own OE content chain. Because the OE content chain of 16100 was calculated before the content chain of 314718 because of its size and therefore has priority, 314718 is referred to as a side chain because it does not actually produce anything new.

Finally, the longest known actual OE content chain follows , namely from 2005020, from which the chain has already been calculated up to the 15163rd position and has also not yet reached an end. At this point you have to factor a 196-digit number.

• 

  OE content chains of the Lehmer Five

• 

  OE content chain of 276

• 

  OE content chain of 552

• 

  OE content chain of 564

• 

  OE content chain from 660

• 

  OE content chain from 966

• 

  OE content chain from 1578 with a record low

• 

  OE content chain of 2340 with the highest number of digits

• 

  OE content chain of 314718 with record length and side chain

• 

  OE content chain from 2005020 with record length without side chain

The next four graphics show special content chains that terminate in a prime number (and then of course end in 1 and then in 0).

First you can see the content chain of 840, which was unsolved for a long time and is therefore part of the Lehmer-Six, but has since been calculated. The chain terminates at the 746th position in the prime number 601 and thus ends at the 748th position in the 0.

The next chain of contents is from the number 19410. This chain starts very steeply and already has an 86-digit number at the 244th position. But then the numbers quickly get smaller again and the chain terminates at the 2200th position in the prime number 43.

Figure 3 shows the content chain of 1638832. This content chain reaches the highest maximum to date, ie the highest number of positions of all previously known terminating content chains. At the 1297th position it reaches the largest 131-digit number that could be factored up to now and at the 3281th position it terminates in the prime number 3. No other terminating content chain has found and factored a higher value.

Figure 4 shows the content chain of 414288. This content chain is currently the longest chain that terminates. It reaches its peak at the 5964th position in a 92-digit number and terminates at the 6584th position in the prime number 601. No other chain is currently longer (and known) that is not an OE chain.

• 

  prime terminated content chain of 840

• 

  Prime terminated content chain from 19410 with a steep rise at the beginning

• 

  Primarily terminated content chain of 1638832 with the highest number of digits

• 

  prime terminated content chain of 414288 with longest chain

Finally, there are a few chains of content that end in a perfect number or a cycle. Since only short content chains with such a property are currently known, the number of digits is not particularly high and therefore hardly differs from one another. So that you get a decent graph , instead of the number of digits, you choose the logarithm of the content number on the y-axis , which always results in the number of digits rounded up.

The content chain from 19362 terminates at the 249th position in the perfect number 8128.

Then you can see the content chain of 976950, which terminates in the 177th position in the perfect number 6.

The next content chains terminate in a cycle.

First you can see the chain of contents of 2856, which ends at the 41st position in the number 14316 and from there merges into a 28 cycle (see example 5 above ).

Then comes the content chain of 9038, which already ends at the 4th position in the number 1184 and from there merges into the 2-cycle 1184/1210.

This is followed by the content chain of 17490, which ends in the 228th position in the number 1264460 and from there merges into a cycle of 4 (1264460/1547860/1727636/1305184).

The next chain of contents is from 18922, which already ends at the 2nd position in the number 12496 and from there goes into a 5-cycle (see example 4 above ).

Finally you can see the content chain of 980460, which ends at the 98th position in the number 2924 and then merges into the 2-cycle 2924/2620. This content chain serves as an example of the fact that chains with a double cycle can of course also be longer.

• 

  The content chain from 19362 ends in the perfect number 8128

• 

  The content chain of 976950 ends in the perfect number 6

• 

  The content chain of 2856 ends in the 28 cycle

• 

  The content chain of 9038 ends in a cycle of two

• 

  The content chain of 17490 ends in a cycle of 4

• 

  The content chain of 18922 ends in a cycle of 5

• 

  The content chain of 980460 ends in a cycle of 2 with a longer chain length

literature

• Eugène-Charles Catalan : A propos d'un théorème de M. Oltramare (December 1887), chapter 294 in Mélanges mathématiques (volume 3), F. Hayez, Brussels 1888, p. 240 (French)
• E. Catalan: Propositions et questions diverses (April 18, 1888), Bulletin de la Société Mathématique de France 16, 1888, pp. 128–129 (French)
• Leonard Eugene Dickson : Theorems and tables on the sum of the divisors of a number , The Quarterly Journal of Pure and Applied Mathematics 44, 1913, pp. 264–296 (English; yearbook review )
• Richard K. Guy : B6. Aliquot sequences and B7. Aliquot cycles. Sociable numbers in Unsolved Problems in Number Theory (3rd edition), Springer-Verlag, New York 2004, ISBN 0-387-20860-7 , pp. 92-97 (English)
• Wolfgang Creyaufmüller: Prime number families - The Catalan problem and the families of prime numbers in the range from 1 to 3000 in detail , ISBN 3-9801032-2-6

Web links

• Eric W. Weisstein : Aliquot Sequence . In: MathWorld (English).
• aliquot sequence - content chain in The Prime Glossary by Chris K. Caldwell
• Prime number families - aliquot sequences - content chains by Wolfgang Creyaufmüller
• Aliquot Sequences - content chains by Paul Zimmermann
• Calculate content chains online
• Current status of aliquot sequences with start term below 3 million - current status of the calculations

Individual evidence

1. ↑ AS Mosunov, What do we know about aliquot sequences?
2. ↑ ^{a b c} Perfect, friendly and sociable numbers. TU Freiberg, accessed on April 23, 2019 . 
3. ↑ content chain of n = 1488
4. ↑ ^{a b} fully calculated content chain of n = 840
5. ↑ ^{a b} Paul Zimmermann: Aliquot sequences 276, 552, 564, 660, 996, 1074 and 1134. Wolfram MathWorld, accessed on August 10, 2017 . 
6. ↑ ^{a b} content chain of n = 276
7. ↑ ^{a b} content chain of n = 552
8. ↑ ^{a b} content chain of n = 564
9. ↑ ^{a b} content chain of n = 660
10. ↑ ^{a b} content chain of n = 966
11. ↑ Rechenkraft.net eV: Current status of aliquot sequences with start term below 3 million. Retrieved April 23, 2019 . 
12. ↑ fully calculated content chain of n = 1248
13. ↑ fully calculated content chain of n = 1848
14. ↑ Juan L. Varona: Aliquot sequences. Retrieved April 23, 2019 . 
15. ↑ Christophe Clavier: Aliquot sequences with leader from 1464 to 9852. Retrieved August 10, 2017 . 
16. ↑ chain of contents of n = 1578
17. ↑ content chain of n = 2340
18. ↑ content chain of n = 314718 (the last 20 known values)
19. ↑ content chain of n = 2005020 (the last 20 known values)
20. ↑ Completed calculated content chain of n = 19410
21. ↑ Completed calculated content chain of n = 1638832
22. ↑ Completed calculated content chain of n = 414288
23. ↑ fully calculated content chain of n = 19362
24. ↑ Completed calculated content chain of n = 976950
25. ↑ fully calculated content chain of n = 2856
26. ↑ completely calculated content chain of n = 9038
27. ↑ fully calculated content chain of n = 17490
28. ↑ fully calculated content chain of n = 18922
29. ↑ Completed calculated content chain of n = 980460