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| What is the Sum of Factors Function? |
The Sum of Factors Function, abreviated s(n) is the sum of all of the proper divisors of a given integer n.
The proper divisors of an integer, n, are all of the positive integers that evenly divide n, including 1, but not including n.
| Example: | s(12) = 1 + 2 + 3 + 4 + 6 = 16 | | |
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| What is an Aliquot Sequence? |
An Aliquot Sequence is a sequence of integers in which each term is the sum of factors of the term before it.
A sequence can begin at any positive integer, and it continues until it terminates, by reaching a prime number, or cycling to a
number already included in the sequence.
| Example: |
The Aliquot sequence starting at 12
s(12) = 16
s(16) = 15
s(15) = 9
s(9) = 4
s(4) = 3 (prime)
12,16,15,9,4,3 |
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| History |
Summing the factors of a number has long been a fascination for mathematicians. The tradition began with Pythagoras and his followers in ancient Greece. They noted that numbers such as 6, 28, and 496 have the special property that the sum of their proper divisors is equal to themselves. (6's proper factors, 1, 2, and 3 sum to 6; 28's factors, 1,2,4,7, and 14 sum to 28...) Because of this property, the Pythagoreans believed that these numbers were perfect, and thus the term "perfect numbers" has been used to describe numbers of this type.
The Pythagoreans were also fascinated by the pair of numbers 220 and 284. In this case each number's factors sum to the other number. (220's factors sum to 284; 284's factors sum to 220.) The Greeks believed that this pair was symbolic of friendship, and we now refer to them as friendly or "amicable numbers." Other mathematicians eventually discovered additional pairs of amicable numbers as well as collections of more than two numbers that cycled, known as sociable numbers
These patterns led to the question of what occurs for the numbers that are not a portion of one of these cycles, what we now know as Aliquot Sequences. It was noted that at least for small numbers the sequences generally terminated relatively quickly. Some sequences are longer and reach fairly high numbers before terminating while a few continue climbing beyond the limits of computation. This led Catalan to propose in 1888 that all sequences would eventually terminate. Recently however empirical evidence has indicated that this may not be true. Guy and Selfridge (1975) conjecture that there are sequences that will never terminate, possibly even the majority of the sequences that begin with even numbers. These conflicting conjectures remain the most important unsolved problem in the study of Aliquot Sequences.
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| Graphing the Sum of Factors Function |
A graph of the sum of factors function is filled with points lying along lines radiating out from near the origin.
Each of these lines contains numbers with similar factorizations, for instance some of the more prominent lines in this chart are the lines that contain all numbers 2 times a prime number, 3 times a prime number or 2*2 times a prime number.
The lines in the graph do not radiate from the origin. The lines themselves are quite predictable, lines with characteristic multiples that do not cotain duplicate factors (square-free numbers) follow this equation:
N = the constant multiple characteristic of the line, (a square-free number)
B = the sum of all of the factors of N including N
x = a value which appears on this line [number which is N times an arbitrary prime]
y = the sum of factors of X |
Lines that do contain duplicate primes will have slight, predictable modifications to take these factors into account.
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