Patterns in Aliquot Sequences And the Sum of Factors Function

1. Patterns in Aliquot Sequences And the Sum of Factors Function By Nathan McNew

2. What is an Aliquot Sequence? An Aliquot Sequence is a sequence in which each term is the sum of the proper factors of the term before it. A number�s proper factors are all of the whole numbers that evenly divide it, other than itself. There is an Aliquot Sequence starting at every positive whole number. The sequence continues until it reaches a prime number, or until it cycles.

3. For Instance: The Aliquot Sequence Starting at 12: 12�s Proper factors are: 1, 2, 3, 4, and 6 which sum to 16. 16�s Proper factor�s 1, 2, 4, and 8 sum to 15 15�s Proper factor�s 1, 3, and 5 sum to 9 9�s Proper factor�s 1, and 3 sum to 4 4�s Proper factor�s 1, and 2 sum to 3 3 is a prime number.

4. The Sum of Factors Function The Sum of Factors function is the function that is repeatedly applied to make an Aliquot Sequence. The result of the Sum of Factors function for a number is found by adding together all of the proper factors of a number.

5. This Sequence can be graphed:

6. Many Sequences Are Similar

10. Types of Aliquot Cycles Perfect Numbers are numbers whose factors sum to themselves. (6; 28; 496) Amicable Numbers are two numbers that sum to each other. (220 � 284) Sociable numbers are a series of more than two numbers whose sums cycle. (12496�14288�15472�14536�14264)

11. An example of a Sequence that ends in an Aliquot Cycle

12. Unterminated Sequences There are some sequences that are not yet known to terminate. There are 5 Sequences under 1000, known as the Lehmer 5 which which are not yet known to terminate. (276, 552, 564, 660, 966) The Catalan Conjecture states that all sequences terminate. The Guy-Selfridge Conjecture states that there are sequences that never terminate, possibly most even numbers.

14. Purpose This Project�s Purpose was to find and analyze patterns in Aliquot Sequences and the Sum of Factors Function. Because these sequences are largely unpredictable, I hoped to find patterns that would help to understand and predict their behavior.

15. Procedure I wrote several C++ programs to calculate Aliquot sequences. These programs used a simple algorithm to find the factors. These programs saved off a variety of statistics and data for analysis.

16. First Term The First Term in each Aliquot Sequence is the Sum of Factors function of that number It is the sum of factors function that defines the aliquot sequences It is necessary to study this function to gain an understanding of how Aliquot Sequences will act.

18. Abundant Numbers Numbers Whose Sum of Factors is greater than themselves Serve to propel sequences upward Make up approximately 24% of all numbers Commonly sum to another abundant number

19. Factorization Numbers with more, small, unique prime factors will be higher on the chart (and more likely to be abundant) For instance, 2^3 *3^2 * 5 * 7 Numbers made up of large factors, or many of the same factors will be lower For instance 51 * 67 or 2^5 * 7

20. Even vs. Odd Numbers will almost always sum to their own class, except: An odd number will sum to an even if it is a perfect square An even number will sum to odd if it is a perfect square or 2 times a perfect square

21. Distribution of the Sum of Factors Function This is the number of times that a value occurs as the sum of factors function of another number. Whenever a number occurs more than once as the sum of factors the aliquot sequences that contain that number merge into one sequence. Once two sequences have merged, they don�t both need to be solved.

33. Conclusions I succeeded in finding numerous patterns in Aliquot Sequences and in the Sum of Factors function. All of these patterns can help us better understand Aliquot Sequences and better predict how they will behave.

34. Further Work I would like to explain: Length Triplets Occurrences of sum of factors function Continue looking for patterns Look into �Drivers� that propel the longest sequences upward