28 Aug 2013

Terminating Primes: Middle Secondary Mathematics Competition Question

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 s0=n, s1=s(n), s2=s(s(n)) and so on, thereby creating the sequence {s0, s1, s2, ...}.

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 s6=7.

c) Prove that it is not possible for a sequence to terminate with a 5, unless s0=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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