29 Aug 2013

A Sequence of Prime Factors: Upper Secondary Mathematics Competition Question

Let f(x) be the sum of the prime factors of the positive integer x, including repeated factors. For example, f(20)=f(2x2x5)=2+2+5=9. Note that f(1)=0 and f(p)=p if p is prime.

Let g(x) be the function g(x)=f(ax+b), where a and b are positive integers. If we iterate g we obtain the sequence g0=x, g1=f(ag0+b) and gn=f(agn-1+b). Such sequences always end in a cycle of length L. Some of these sequences terminate at a fixed point P with a cycle length of 0.

For example, if g(x)=f(3x+1) and x=14 we get the sequence {14, 43, 20, 61, 29, 17, 17} which terminates at the fixed point 17.

Let's look specifically at the function g(x)=f(5x+3).

a) Calculate the sequence generated from x=40 and find its value of L.

b) Find the two fixed points of this sequence that are both less than 100.

Note that such sequences have not been exhaustively analysed and there are a number of open questions that I will discuss in another post.

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