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A039654
a(n) = prime reached by iterating f(x) = sigma(x)-1 starting at n, or -1 if no prime is ever reached.
21
2, 3, 11, 5, 11, 7, 23, 71, 17, 11, 71, 13, 23, 23, 71, 17, 59, 19, 41, 31, 47, 23, 59, 71, 41, 71, 71, 29, 71, 31, 167, 47, 53, 47, 233, 37, 59, 71, 89, 41, 167, 43, 83, 167, 71, 47, 167, 167, 167, 71, 97, 53, 167, 71, 167, 79, 89, 59, 167, 61, 167, 103, 311, 83, 167, 67
OFFSET
2,1
COMMENTS
It appears nearly certain that a prime is always reached for n>1.
Since sigma(n) > n for n > 1, and sigma(n) = n + 1 only for n prime, the iteration either reaches a prime and loops there, or grows indefinitely. - Franklin T. Adams-Watters, May 10 2010
Guy (2004) attributes this conjecture to Erdos. See Erdos et al. (1990). - N. J. A. Sloane, Aug 30 2017
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 149.
LINKS
Franklin T. Adams-Watters, Table of n, a(n) for n=2..10000
Lucilla Baldini and Josef Eschgfäller, Random functions from coupled dynamical systems, arXiv preprint arXiv:1609.01750 [math.CO], 2016. Mentions the conjecture.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
MATHEMATICA
f[n_]:=Plus@@Divisors[n]-1; Table[Nest[f, n, 6], {n, 2, 5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2010 *)
f[n_] := DivisorSigma[1, n]-1; Table[FixedPoint[f, n], {n, 2, 100}] (* T. D. Noe, May 10 2010 *)
PROG
(PARI) a(n)=local(m); if(n<2, 0, while((m=sigma(n)-1)!=n, n=m); n) \\ Franklin T. Adams-Watters, May 10 2010
(PARI) A039654(n)=n>1&&until(n==n=sigma(n)-1, ); n \\ M. F. Hasler, Sep 25 2017
CROSSREFS
Cf. A039655 (the number of steps needed), A039649, A039650, A039651, A039652, A039653, A039656, A291301, A291302, A291776, A291777.
For records see A292112, A292113.
Cf. A177343: number of times the n-th prime occurs in this sequence.
Cf. A292874: least k such that a(k) = prime(n).
Sequence in context: A084743 A030391 A244496 * A075240 A347358 A333200
KEYWORD
nonn
EXTENSIONS
Contingency for no prime reached added by Franklin T. Adams-Watters, May 10 2010
Changed escape value from 0 to -1 to be consistent with several related sequences. - N. J. A. Sloane, Aug 31 2017
STATUS
approved