Let s(n) be the sum of the

*proper*factors of a positive integer n; this is the sum of all the factors of n, including 1 but excluding n itself. Let s

_{0}=n, s

_{1}=s(n), s

_{2}=s(s(n)) and so on, thereby creating the sequence {s

_{0}, s

_{1}, s

_{2}, ...}.

If n is a prime number p, then s(p)=1 and s(s(p))=0, thus terminating the sequence. As most such sequences terminate in this way, it is normal to terminate the sequence at the first prime number.

a) Calculate the terminating prime number for the starting value of n=12.

b) Find all possible sequences such that s

_{6}=7.

c) Prove that it is not possible for a sequence to terminate with a 5, unless s

_{0}=5.

These types of sequences are still being researched and they do not all terminate in the manner described above. Try n=276 and see what happens. Have fun!

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