An aliquot sequence can end in one of four ways:
n Divisors Sum 12 1, 2, 3, 4, 6 16 16 1, 2, 4, 8 15 15 1, 3, 5 9 9 1, 3 4 4 1, 2 3 3 1 1 1 end of the sequence .
An example of the third case is 220:
n Divisors Sum 25 1, 5 6 6 1, 2, 3 6
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. Wolfgang Creyaufmueller is 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.
n Divisors Sum 220 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 284 284 1, 2, 4, 71, 142 220
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) ]The sum of the aliquot divisors is then 28 - 12 = 16.
= [ (8 - 1) / (2 - 1) ] * [ (9 - 1) / (3 - 1) ]
= [ 7/1 ] * [ 8/2 ] = 7 * 4 = 28
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 * 653Click here to see all of terms (up to index 876) in the sequence.
The sequence has been computed up to the 876th term, which is:
#876 = 9408615261469505887706671750793624823866119536939826913025961862347\
5379527031440042131694744559472874039813661586792360539867881463364\
5190331285746834115670010914788678164904
= 2^3 * 3 * 28807 * 3715849908970517
* 36623367227260719942192368260721274640565915679942696857613631195\
76966981222962844456903517464712587352119799328815870854957796942\
17712253893620881894109
Largest term has 174 digits at index 876
The last factor above (3662336...1894109) is a 153-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 June 11, 2011.