A292112 -id:A292112 - cp's OEIS frontend

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

David W. Wilson

nonn

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

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 149.

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)

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).

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 *)

a(n)=local(m);if(n<2,0,while((m=sigma(n)-1)!=n,n=m);n) \\ Franklin T. Adams-Watters, May 10 2010

A039654(n)=n>1&&until(n==n=sigma(n)-1,);n \\ M. F. Hasler, Sep 25 2017

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

A292113 List of numbers n such that A039654(n) reaches a new record high.

Original entry on oeis.org

2, 3, 4, 8, 9, 32, 36, 64, 81, 100, 121, 144, 228, 256, 300, 400, 441, 468, 800, 1200, 2964, 5202, 5408, 6084, 6400, 7500, 8100, 9216, 24556, 28092, 31329, 32176, 32400, 37296, 49017, 49152, 57600, 72156, 80400, 83161, 86352, 88200, 133200

Offset: 1

N. J. A. Sloane, Sep 22 2017

nonn

Naively, one might have expected these numbers to have some other distinguishing property (primorials, perhaps), but that seems not to be the case.

Except for 3 of the listed terms, a(n)-1 or a(n)+1 has at most 2 prime divisors. The same is true for many of the terms themselves, often of the form 2^k, 3^k, 2^k*3^k' or 2^k*5^k'. (Factorization of the terms: 2, 3, 2^2, 2^3, 3^2, 2^5, 2^2*3^2, 2^6, 3^4, 2^2*5^2, 11^2, 2^4*3^2, 2^2*3*19, 2^8, 2^2*3*5^2, 2^4*5^2, 3^2*7^2, 2^2*3^2*13, 2^5*5^2, ...) - M. F. Hasler, Sep 25 2017

Hugo Pfoertner, Table of n, a(n) for n = 1..56

Cf. A039654, A039655, A292112.

m=-n=1; until(print1(n","), until(A039654(n++)>m,); m=A039654(n)) \\ M. F. Hasler, Sep 25 2017

A177343 Number of times the n-th prime occurs in A039654.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 4, 2, 4, 1, 1, 12, 1, 2, 3, 3, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 3, 1, 1, 25, 1, 4, 2, 10, 1, 1, 1, 1, 3, 5, 1, 4, 10, 1, 7, 1, 8, 3, 2, 1, 1, 1, 4, 2, 5, 1, 1, 1, 1, 1, 1, 1, 18, 1, 1, 1, 10, 2, 1, 1, 1, 6, 1, 16, 4, 2, 2, 3, 1, 1, 1, 3, 11, 1, 2, 1, 18, 1, 2, 1, 1, 1, 3

Offset: 1

Franklin T. Adams-Watters, May 06 2010

nonn

Record values for primes up to 10000:
    n  p(n) a(n)
  2   1
 11   3
 23   4
 71  12
167  25
743  58
1487  62
2879  71
5039 125
6719 168
The function A039653(n) = sigma(n)-1 iterated in A039654 satisfies A039653(n) >= n (with equality iff n is a prime), therefore the prime p cannot appear beyond index p in A039654, and it is sufficient to count how many times p = A039654(n) with n < p, cf. Formula. - M. F. Hasler, Sep 25 2017

Franklin T. Adams-Watters, Table of n, a(n) for n=1..1229 (primes through 10000)

Cf. A039654, A039653, A292112, A292113.

a(n)=sum(k=2,n=prime(n),A039654(k)==n) \\ M. F. Hasler, Sep 25 2017

a(n) = 1 + # { k < prime(n) | A039654(k) = prime(n) } . - M. F. Hasler, Sep 25 2017

A292874 Least k such that A039654(k) = prime(n).

Original entry on oeis.org

2, 3, 5, 7, 4, 13, 10, 19, 8, 29, 21, 37, 20, 43, 22, 34, 18, 61, 67, 9, 73, 57, 44, 40, 52, 101, 63, 85, 109, 74, 93, 86, 137, 76, 149, 91, 157, 163, 32, 173, 88, 117, 68, 193, 197, 199, 211, 84, 80, 229, 36, 104, 241, 96, 257, 102, 136

Offset: 1

M. F. Hasler, Sep 25 2017

nonn

We have A039653(k) >= k, thus also A039654(k) >= k, with equality if k is prime, therefore a(n) <= prime(n), and the largest k for which A039654(k) = prime(n) is always k = prime(n).

Cf. A039653, A039654, A177343, A292112, A292113.

a039654(n) = if(n<2, 0, my(m); while((m=sigma(n)-1)!=n, n=m); n);
a(n) = my(k=0, p=prime(n)); while(a039654(k) != p, k++); k; \\ Michel Marcus, Sep 26 2017

A292876 Irregular table whose n-th row lists all k such that A039654(k) = prime(n).

Original entry on oeis.org

2, 3, 5, 7, 4, 6, 11, 13, 10, 17, 19, 8, 14, 15, 23, 29, 21, 31, 37, 20, 26, 41, 43, 22, 33, 35, 47, 34, 53, 18, 24, 38, 59, 61, 67, 9, 12, 16, 25, 27, 28, 30, 39, 46, 51, 55, 71, 73, 57, 79, 44, 65, 83, 40, 58, 89

Offset: 1

M. F. Hasler, Sep 25 2017

This can also be considered as a list of all orbits of A039653, ordered by their maximal element p = A039654(k) for any k of this orbit.
Indeed, A039653(x) >= x with equality iff x is prime, and all orbits of A039653 are conjectured to end in such a fixed point prime p = A039654(k) for any k in this orbit.
Row lengths are given by A177343.
This sequence is also a permutation of all integers > 1, where each prime p(k) is immediately preceded by A177343(k)-1 composite numbers less than p(k). It follows that each composite is preceded either by a smaller composite or by a larger prime, and followed by a larger composite or prime. Thus, the primes appear in their natural order, but the composites do not.
The first element of each row (i.e., the first column of this table) is given by A292874.
We see (cf. a-file) that powers of 2 are often the first element (or at least part) of relatively long orbits: A177343(A000720(A039654(2^k))) = (1, 3, 4, 12, 25, 5, 10, 35, 61, 143, 143, 220, 365, ...)

			The table starts:
    n  p(n)  { k | A039654(k) = p(n) }
    1    2   { 2 }
    2    3   { 3 }
    3    5   { 5 }
    4    7   { 7 }
    5   11   { 4, 6, 11 }
    6   13   { 13 }
    7   17   { 10, 17 }
    8   19   { 19 }
    9   23   { 8, 14, 15, 23 }

M. F. Hasler, Table rows n = 1..1229

Cf. A039654, A039653, A177343, A292874, A292112, A292113.

A292876(n,p=prime(n))=select(k->A039654(k)==p,[2..p]) \\ Not optimized nor efficient; mainly for illustrational purpose. - M. F. Hasler, Sep 25 2017

cp's OEIS Frontend

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

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A292113 List of numbers n such that A039654(n) reaches a new record high.

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A177343 Number of times the n-th prime occurs in A039654.

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A292874 Least k such that A039654(k) = prime(n).

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A292876 Irregular table whose n-th row lists all k such that A039654(k) = prime(n).

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