Aliquot sequence

From FMNWiki
Jump to: navigation, search

An aliqout sequence is a sequence of numbers where the next number in the sequence is the sum of the proper divisors of the previous term.

For example, the aliquot sequence beginning with 12 is as follows:

12, 16, 15, 9, 4, 3, 1, 0

  • The proper divisors of 12 are 6, 4, 3, 2, and 1; 6+4+3+2+1=16
  • The proper divisors of 16 are 8, 4, 2, and 1; 8+4+2+1=15
  • The proper divisors of 15 are 5, 3, and 1; 5+3+1=9
  • The proper divisors of 9 are 3 and 1; 3+1=4
  • The proper divisors of 4 are 2 and 1; 2+1=3
  • The only proper divisor of 3 is 1
  • There are no proper divisors of 1, so the sequence is terminated with a 0

I'll represent entire sequences as An where n is the initial term. I'll represent individual terms of the sequences as ai, where a0 = n. To represent the ith term of An, I'll use the convention An:i.

Contents

The Sum-of-Divisors function

The sum-of-positive-divisors function, σx(i), valid for any value of x (real or complex) is the sum of the xth power of all the divisors of i.

σx(i) = ∑dx
d | i

A "count" of the divisors of n can be obtained when x=0. For instance:

σ0(12) = ∑d0 = 120 + 60 + 40 + 30 + 20 + 10 = 1 + 1 + 1 + 1 + 1 + 1 = 6
d | 12

And the sum of these divisors is obtained with x=1:

σ1(12) = ∑d1 = 121 + 61 + 41 + 31 + 21 + 11 = 12 + 6 + 4 + 3 + 2 + 1 = 28
d | 12

An aliquot sequence can now be thought of in the following way:

An:i + 1 = ai + 1 = σ1(ai) − ai

Data

I ran across this sequence by accident one day, and was curious as to certain aspects of its properties. For instance, how long is a given sequence? Some inital data points are given below:

a0 sequence length
1 2
2 3
3 3
4 4
5 3
6 \infty
7 3
8 4
9 5
10 5

This is obviously just a taste, but some conclusions are easily drawn from the construction of the sequence so only the more interesting sequences emerge.

  • Because a prime number phas only one proper divisor, 1, any sequence beginning with a prime number will always be p,1,0.
  • Because a perfect number is by definition the sum of its proper divisors, any sequence beginning with a perfect number will always be an infinite repetition of this term.
  • Sequences converge: the second term of A10, or A10:2 = 8. Thus the remainder of the sequence is equivalent to A8.
  • A number following any 2n in this sequence will always be 2n − 1. I'll leave the proof of this up to the reader (trust me, it's an easy one).

The truly interesting sequences are those which contain repeating sequences with length greater than one. The first of these is A220: 220, 284, 220, 284, etc.

  • If the period is 2, the numbers that make up this period are called an "amicable pair".
  • If the period is greater than 2, the numbers that make up the period are called "sociable numbers".
  • Those numbers whose sequence converges to a perfect, amicable, or sociable number are called "aspiring numbers"

It is as yet unknown if any sequence is infinite and aperiodic, though there are several candidates for those properties. As of October 2009 (according to Wikipedia), there are 9376 numbers less than 1000000 whose aliquot sequences have not been fully determined.

The first five, known as the Lehmer Five, are 276, 552, 564, 660, and 966. The group was once known as the Lehmer Six, until 840 dropped out when the sequence was "closed" with 749 terms.

Interesting Sequences

In the following tables, the nomenclature used in the "result" column is how the sequence Anends, or converges:

  • an "f" indicates that the number is "from" another sequence. The numbers x, y that follow indicate that the number is Ax:y
  • an "m" indicates that the sequence merges with another. The numbers x, y that follow indicate that the sequence An merges with Ax at Ax:y
  • an "x" indicates that the number is the root of a sequence.

Examples:

  • The table indicates 828 has a "result" of "m 660, 2, 2". This means that A828:2 = A660:2.
  • The table indicates 696 has a "result" of "f 276, 3". This means that A276:3 = 696.

I will only be listing results here up to the number 10000. I'll serve up files at some later date with more numbers.

Open-Ended

The following numbers have sequences that are not known to terminate. Most sites list only the primitives, (those with "x" in the "result" column); I'm listing all of them.

n result
276 x
306 m 276, 2, 2
396 f 276, 2
552 x
564 x
660 x
696 f 276, 3
780 f 564, 2
828 m 660, 2, 2
888 f 552, 2
966 x
996 m 660, 2

Perfect Numbers

These are numbers that will repeat themselves forever in an aliquot sequence. 47 are known, and all known perfect numbers correspond directly to Mersenne primes.

Perfect number First appearance
6 6
28 28
496 496
8128 8128

Amicable Pairs

The following numbers are amicable pairs. For brevity I will not list sequences that converge to the pairs, just the pairs themselves. Likewise, I'll only put a pair in the table once: 220/284 will not show up as 284/220. The sequences here alternate between the two numbers. The "First appearance" lists the sequence number

x y First appearance
220 284 220
1184 1210 1064
2620 2924 1188
5020 5564 5020
6232 6368 6232

Sociable Numbers

These numbers make up "cycles" in the aliquot sequence. Amicable numbers are a class of sociable numbers, with a cycle length of two. Those listed below are longer cycles, along with their first appearance.

cycle length First appearance
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 28 2856
12496, 14288, 15472, 14536, 14264 5 9464


More Than 100 Terms

If the sequence terminates, but still contains more than 100 terms (again, only those to 1000 for the time being), I've listed those in this table, along with the number of terms in that sequence.

n terms
138 179
150 178
168 176
222 177
234 176
312 175
528 174
570 174
702 302
720 196
726 174
840 749
858 169
870 173
936 186
960 173
978 301
990 300
Retrieved from "http://findingmyname.com/wiki/index.php/Aliquot_sequence"
Personal tools