Aliquot sequence
Under a content chain ( also aliquot sequence of engl. Aliquot sequence) refers to a sequence of positive integers, in which each of the number of content (the sum of the proper divisors ) of its predecessor. For example, (10, 8, 7, 1, 0), the contents of the chain of 10:
Natural numbers that lead over the content chains in the same prime number (except for the 0 and 1 ), which form called a prime family (English prime family), shortly also P family (german p -family). One Ring Family (german motorcycle family), shortly known as R- Family (german c -family), terminated in a ring perfect, of friendly and sociable numbers.
The Catalan - Dickson conjecture (named after Eugène Charles Catalan and Leonard Eugene Dickson ) says that any content chain is periodically or ends with 0. Until today it is neither proved nor disproved.
The lengths of the contents of tracks for n = 1, 2, ... other than the starting value, are 1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, ... ( sequence A044050 in OEIS ).
The contents of chains of 25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, ... ( sequence A063769 in OEIS ) terminate in a perfect number.
Chains of the content 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ... ( in sequence A080907 OEIS ) terminate in the 0.
Content chains can be generated, for example in the factoring database.
Formal definition
The content chain with the start value n or content chain of n is the consequence
Where the divisor sum.
Lehmer and Six -Five
The first six open ( not fully calculated ) chains in the interval [ 1, 1000] were named after the couple Derrick Lehmer and Emma Lehmer, Lehmer -Six. Start your numbers were 276, 552, 564, 660, 840 and 966
The chain with the starting number 840 is now fully known. They terminated in the prime number 601, followed by 1 and 0 The remaining 5 open chains Lehmer Five are now called. The chain with the starting number 276 was to s1661 (276 ) = 828579689207854231700104280392137451390009094307100079055497138748077003429376547829092101844491763309481122250886156621164877126867618972297907847860949694119802 calculated. Apart from 2, 32 and 17 no other prime factors are more known. The chain with the starting number 552 is up to s1000 (552 ) = 2794872109104142939547040872964671736140257182466402607234754288301809152153432092781830003844507869549816694690620457413683217553432931111904090024134787039805079744 = 26 · 32 · 23 · 3386311 ·? known, the chain with the starting number 564 up to s3331 (564 ) = 62480132633545725517909281438721367091842001184520349840724510325207881275036412684818098341758828567208076019524744242876290219121978072647629225267249508248342007605868 = 22 · 32 · 7 · 83 ·? , The chain with the starting number 660 to S868 (660 ) = 16788048498108629410245989258167410533133694355609925867939682790692192589086073968208670999907251359882905744795251057874406030687132867210822582738876902402909209285299640 = 23 · 33 · 5 · 130 241 801 411 · · 16005149345444515675217297? and the chain with the starting number 966 to S830 ( 966 ) = 416972214461346415624349922965959741224644564312867092205292393763956854903020121271060952565951183457861634303355328940722929059955235489819461527450056524348602344 = 23 · 3 · 61 ·? .
There is also in the interval [ 1, 10000] currently 81 open chains in [1, 100000 ] 902 and [1, 106 ] 9312th For these chains, no name has prevailed.