CTK Exchange Front Page Movie shortcuts Personal info Awards Reciprocal links Terms of use Privacy Policy Cut The Knot! MSET99 Talk Games & Puzzles Arithmetic/Algebra Geometry Probability Eye Opener Analog Gadgets Inventor's Paradox Did you know?... Proofs Math as Language Things Impossible My Logo Math Poll Other Math sit's Guest book News sit's Recommend this site        CTK Exchange

 Subject: "sum of prime factors and cycles" Previous Topic | Next Topic
 Conferences The CTK Exchange This and that Topic #708 Printer-friendly copy Email this topic to a friend Reading Topic #708
erszega
Member since Apr-23-05
May-17-06, 06:41 AM (EST)    "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):810,14,9,2310,10,9,19,1139,43,43,45,17,33,12,11,23,19,12,31,43.Interestingly, there seem to be similar results when taking the sum of prime factors of more than two preceding terms. For instance, withF(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.

 Conferences | Forums | Topics | Previous Topic | Next Topic
 Select another forum or conference Lobby The CTK Exchange (Conference)   |--Early math (Public)   |--Middle school (Public)   |--High school (Public)   |--College math (Public)   |--This and that (Public)   |--Guest book (Public) Educational Press (Conference)   |--No Child Left Behind (Public)   |--Math Wars (Public)   |--Mathematics and general education (Public) You may be curious to have a look at the old CTK Exchange archive.  