A039654 a(n) = prime reached by iterating f(x) = sigma(x)-1 starting at n, or -1 if no prime is ever reached.

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

Author

David W. Wilson

Keywords

• nonn

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

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).

Programs

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 *)

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

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