[seqfan] Re: Iterating sigma - 1

Robert G. Wilson v rgwv at rgwv.com
Fri May 7 05:49:28 CEST 2010


    You wish to submit a sequence of the least k such that the number of 
iterations is n?


From: "T. D. Noe" <noe at sspectra.com>
Sent: Thursday, May 06, 2010 5:46 PM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Iterating sigma - 1

> At 4:51 PM -0400 5/6/10, franktaw at netscape.net wrote:
>>Consider http://www.research.att.com/~njas/sequences/A039654 - Prime
>>reached by iterating f(x) = sigma(x)-1 on n.
>>It isn't obvious that this iteration always reaches a prime, although
>>it seems nearly certain that it does. Should we add something like ",
>>or 0 if no prime is ever reached", with a comment that apparently the
>>sequence always does reach a prime? (Or can someone prove that a prime
>>is always reached?)
>>One might, in this case, also add a(1) = 0 (suitably modifying the
> For n>1, the iteration "x=n, repeat x=f(x) until a fixed point is reached"
> will either increase indefinitely or converge to a prime.  There is no
> possibility of looping because f(x) >= x for all x>1.  If the iteration
> converges, it will converge to a prime because sigma(x)=x+1 iff x is 
> prime.
> The following Mma code is better because it doesn't stop after 6 
> iterations:
> f[n_] := DivisorSigma[1,n]-1; Table[FixedPoint[f,n], {n,1000}]
> The iteration converges for all 1 < n <= 10^6. In that range, the maximum
> number of iterations is 45.
> Tony
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