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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. 18
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 (list; graph; refs; listen; history; text; internal format)
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, 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]

Math Overflow discussion, Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

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 A229607 A137332

Adjacent sequences:  A039651 A039652 A039653 * A039655 A039656 A039657

KEYWORD

nonn,changed

AUTHOR

David W. Wilson

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

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Last modified September 26 10:38 EDT 2017. Contains 292518 sequences.