Aliquot Sequences

This is my page about aliquot sequences. Some others who have pages about aliquot sequences are:

Introduction

The aliquot divisors of a number are all of its divisors except itself. So, for example, the aliquot divisors of 12 are 1, 2, 3, 4, and 6. The aliquot divisors of 10 are 1, 2, and 5. An aliquot sequence is a sequence of numbers where each number is the sum of the previous number’s aliquot divisors. For example, the aliquot sequence starting at 12 is: 12, 16, 15, 9, 4, 3, 1. It is derived as follows:
nDivisorsSum
121, 2, 3, 4, 616
161, 2, 4, 815
151, 3, 59
91, 34
41, 23
311
1end of the sequence.
An aliquot sequence can end in one of four ways:
  1. with a prime number, followed by 1, as above.
  2. with a perfect number, such as 6.
  3. with a cycle of two or more numbers.
  4. increases without limit.
An example of the second case is 25:
nDivisorsSum
251, 56
61, 2, 36
An example of the third case is 220:
nDivisorsSum
2201, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 284
2841, 2, 4, 71, 142220
It is currently not known whether any number leads to a sequence that increases without limit (case 4). In an attempt to decide this question, many aliquot sequences have been computed. The ultimate fate of the sequences for all numbers up to 1000 has been found to be one of the first three cases, except for 276, 552, 564, 660, and 966. The ends of these five sequences (the “Lehmer five”) are currently unknown. Various people, including most of those listed at the top of this page, are working on sequences with larger starting values. I have done some work on the sequence that starts with 966. See below for the status of this sequence.

Computing the Sum of Divisors

This part of the page uses superscriptslike this. If your browser does not display superscripts, the formulas below will not be shown correctly.

It is not actually necessary to list all of the divisors of a number to compute the sum of the divisors. Instead, there is a formula that uses the prime factorization of the number. If the factorization of N is

N = pa * qb * rc * ...
where p, q, r, etc. are the prime factors, and a, b, c, etc. are the corresponding exponents, then the sum of the divisors is:
Sum = [ (p(a+1) - 1) / (p - 1) ] * [ (q(b+1) - 1) / (q - 1) ] * [ (r(c+1) - 1) / (r - 1) ] * ...
This Sum includes the number N itself as a divisor, so we need to subtract N from it to get the sum of the aliquot divisors.

As an example: 12 = 22 * 31 (usually written: 22 * 3). The sum of its divisors (including 12 itself) is:

Sum = [ (23 - 1) / (2 - 1) ] * [ (32 - 1) / (3 - 1) ]
= [ (8 - 1) / (2 - 1) ] * [ (9 - 1) / (3 - 1) ]
= [ 7/1 ] * [ 8/2 ] = 7 * 4 = 28
The sum of the aliquot divisors is then 28 - 12 = 16.

The aliquot sequence for 966

The first several numbers in the aliquot sequence for 966 are:
Starting value: 966

#0 = 966 = 2 * 3 * 7 * 23
#1 = 1338 = 2 * 3 * 223
#2 = 1350 = 2 * 3^3 * 5^2
#3 = 2370 = 2 * 3 * 5 * 79
#4 = 3390 = 2 * 3 * 5 * 113
#5 = 4818 = 2 * 3 * 11 * 73
#6 = 5838 = 2 * 3 * 7 * 139
#7 = 7602 = 2 * 3 * 7 * 181
#8 = 9870 = 2 * 3 * 5 * 7 * 47
#9 = 17778 = 2 * 3 * 2963
#10 = 17790 = 2 * 3 * 5 * 593
#11 = 24978 = 2 * 3 * 23 * 181
#12 = 27438 = 2 * 3 * 17 * 269
#13 = 30882 = 2 * 3 * 5147
#14 = 30894 = 2 * 3 * 19 * 271
#15 = 34386 = 2 * 3 * 11 * 521
#16 = 40782 = 2 * 3 * 7 * 971
#17 = 52530 = 2 * 3 * 5 * 17 * 103
#18 = 82254 = 2 * 3 * 13709
#19 = 82266 = 2 * 3 * 13711
#20 = 82278 = 2 * 3^2 * 7 * 653
Click here to see all of terms (so far) in the sequence.

The sequence has been calculated up to the 959th term, which is:

#959 = 75521278771217953409751209776054073314771137357812063153906615114\
       56191440344410174143346010432586733455633829814351208029335820367\
       94114579306773775418042614407002005671675856948450923010096

     = 2^4 * 3^3 * 23 * 29
       * 262095614592765955250677472985917018278260652166319837143603944\
         953085659959756586086933825116351086035302967606972597313472979\
         4952503312855054326223756325359566751366232424957481158459
The last factor above (2620956...1158459) is a 184-digit composite number, whose factors are currently unknown.

Usually, each term in the sequence is larger than the preceding term. However, each of the terms #658, #659, and #660 is smaller than the preceding term. Then the sequence increases again, starting with term #661. The sequence decreases again from #704 thru #708.


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Last updated on May 10, 2017.