27 Aug 2013

An Odd Sequence: Middle Secondary Mathematics Competition Question

A function T(x) is defined as follows, where x is a positive integer:

If x is even, divide x by 2;

If x is odd, calculate the sum of all the factors of x (including 1 and x itself).

Repeat these rules, thereby creating a sequence of numbers. Let S0=x, S1=T(x), S2=T(T(x)) and so on, so that Sn=Tn(x). Also, let m be the first iteration at which Sm=Tm(x)=1. Note that T(1)=1, so we terminate the sequence at the first 1 we encounter.

For example, if x=5, S0=5, S1=6, S2=3, S3=4, S4=2, S5=1. This results in the sequence {5, 6, 3, 4, 2, 1}, so that for x=5, m=5.

a) Find the value of m for x=121.

b) Find the values of x for which m=7.

One open question to ponder is whether such a sequence terminates for every starting value of x.

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