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Subject: "sum of prime factors and cycles"     Previous Topic | Next Topic
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erszega
Member since Apr-23-05
May-17-06, 06:41 AM (EST)
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"sum of prime factors and cycles"
 
   Let sopfr(n) be the sum of prime factors of n (see https://mathworld.wolfram.com/SumofPrimeFactors.html ).

There have been observations that certain iterations involving the sopfr function seem to lead invariably to cycles, or closed loops (see for instance https://www.mathpages.com/home/kmath006/part6/part6.htm ).

I looked at sequences of the form F(n)=sopfr(F(n-1)+F(n-2)), where F(1) and F(2) are any positive integers. These sequences also appear to end in repeating cycles sooner or later, in fact I have found the following four cycles (of length 1, 4, 5, and 13):
8
10,14,9,23
10,10,9,19,11
39,43,43,45,17,33,12,11,23,19,12,31,43.

Interestingly, there seem to be similar results when taking the sum of prime factors of more than two preceding terms. For instance, with
F(n)=sopfr(F(n-1)+F(n-2)+F(n-3)+F(n-4)+F(n-5)), a possible cycle is one that has a length of 31196 terms. You can start that cycle with, for instance,
49,93,435,98,92.

With F(n)=sopfr(F(n-1)+...+F(n-7)), a possible cycle has a length of 28274564 (if my computation was right).

Again similar results can be obtained by checking sequences of the type F(n) = sopfr(F(n-1)*F(n-2)) = sopfr(F(n-1)) + sopfr(F(n-2)), and so on.

Cycles also appear when using the sopfr function with cellular automata. Eg, taking an elementary (one dimensional) cellular automaton, wich is circular or finite ( with n sit's, site(n+1) = site(1) ), and a rule that computes the next value of a site as the sum of the prime factors of the values of specified neighbouring sites, the subsequent sums of the values of all n sites will also form a sequence ending in a repeating cycle.


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