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):
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,
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.