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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

578.0. "Non-factor Series" by CLT::GILBERT (eager like a child) Sun Sep 14 1986 01:41

(a)  Let f(n) be the smallest positive integer which is not a factor
     of n.  Continue the series f(n), f(f(n)), f(f(f(n))), ... until
     you reach 2.  What is the maximum length of the series?

(b)  Let g(n) be the second smallest positive integer which is not a
     factor of n.  Continue the series g(n), g(g(n)), g(g(g(n))), ...
     until you reach 2.  What is the maximum length of the series?


[This problem is due to David Grabiner of Claremont H.S., Claremont, CA]
T.RTitleUserPersonal
Name
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578.1Answer to (a).CHOVAX::YOUNGI think we're all bozos on this BUS.Sun Sep 14 1986 17:2725
    (a):
    
    	The maximum length of such a series is 3.
    
    	The smallest postive integer which is not a factor of n, must
    be of the form:
    			 m
    			P
    				Where (P) is any prime, and (m) is any
    natural number.  Now if (n) was 2 then we were done before we started,
    so that comprises 0 steps.  If (n) is any odd number then f(n)=2
    so that we have 1 step.  If (n) is any even number not divisible
    by 3 then f(n)=3, f(f(n))=2, for 2 steps.  All remaining (n) must
    then be of the form:
    				 a   b
    				2 * 3 * ...
    	      m					Now f(n) must be of
    the form P  <> (2,3).  Once again, if P is odd then f(f(n))=2
    for 2 steps.  However, if P is 2 then f(f(n))=3, and f(f(f(n)))=2
    which is 3 steps.
    
    This covers all possibilities.
                  
    -- Barry
    
578.2Answer to (b).CHOVAX::YOUNGI think we're all bozos on this BUS.Sun Sep 14 1986 17:335
    (b):
    	This series can NEVER reach 2.
    
    	Reason:  2 is never the second smallest non-factor of any number.
    
578.3CLT::GILBERTeager like a childTue Sep 16 1986 21:251
(c)  Same as (b), but continue until you reach 3.