Dan Asimov's misinterpretation of Propp's notion of "squarefree part" is still a perfectly good problem. If we redo my probabilistic heuristic analysis for that problem: can Propp/Asimov's sequences f[n] = CorePart( f[n-1]+f[n-2] ) ever (with suitable starting integers) diverge to infinity without instead falling into a finite length cycle? Here CorePart(X) means the product of all the primes inside X's factorization, e.g. CorePart( 2*3*3*5*7*7*7*7 ) = 2*3*5*7. (Is that what Asimov meant?) If we model the X as "random" then with chance 1/4 they contain a factor of 2 that will be removed, with chance 1/8 a fatcor 4 is removed, chnace 1/9 a factor 3 is removed, etc. So ln(X) is going to decrease due to these removals, by an expected additive amount DELTA with DELTA <= SUM_{k>1} SUM_{q>0} k^(-1-q) * ln(k^q) = 1.77 The expected decrease amount also is lower bounded by the same sum but summed over prime k only and demanding q=1 only: DELTA > 0.49. Also note, the "probability distribution" of DELTA has finite variance. Meanwhile, ln(X) is going to grow due to the Fibonacci-like recurrence, by some amount presumably less than ln(2) = 0.693147180559945 in "expectation." So... it looks unclear whether Asimov's problem-version can go infinite and avoid a cycle, in this probability model. Need less-sloppy upper and lower bounds on DELTA. It seems to be a close thing, when I try to be less sloppy; e.g. one of my less-sloppy lower bounds was 0.642 which is quite close to 0.693. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)