A291302 -id:A291302 - 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

A291301 a(n) = prime that is eventually reached when x -> sigma(x)-1 is repeatedly applied to the product of the first n primes, or -1 if no prime is ever reached.

Original entry on oeis.org

2, 11, 71, 743, 6911, 117239, 2013983, 34836479, 921086711, 33596203871, 18754852859999, 1306753691335679, 2795529813471359, 200489563747397471, 7143750592470475271, 146095655504943513599, 161739770170976834876927, 543475838478389870591999, 317180662337566737324195839

Offset: 1

N. J. A. Sloane, Aug 31 2017

nonn

A subsequence of A039654.

			2*3*5*7*11*13 = 30030 -> 96767 -> 111359 -> 117239 takes three steps to reach a prime, so a(6) = 117239.

Chai Wah Wu, Table of n, a(n) for n = 1..35

Cf. A039654, A000203, A002110, A291302 (number of steps).

p[n_]:=Times@@Prime/@Range[n]; f[n_]:=DivisorSigma[1,n]-1;
a[n_]:=Last[NestWhileList[f,p[n],CompositeQ]]; a/@Range[20] (* Ivan N. Ianakiev, Sep 01 2017 *)

from sympy import primorial, isprime, divisor_sigma
def A291301(n):
    m = primorial(n)
    while not isprime(m):
        m = divisor_sigma(m) - 1
    return m # Chai Wah Wu, Aug 31 2017

a(10)-a(19) from Chai Wah Wu, Aug 31 2017

A291776 a(n) = prime that is eventually reached when x -> sigma(x)-1 is repeatedly applied to 2^n-1, or -1 if no prime is ever reached.

Original entry on oeis.org

3, 7, 23, 31, 103, 127, 431, 911, 1847, 6719, 10487, 8191, 56999, 41399, 135647, 131071, 560159, 524287, 1999871, 3982271, 5909759, 17512991, 46092239, 46335599, 164460119, 186592247, 736727807, 3926707199, 4146049487, 2147483647, 8994904463, 11132323439

Offset: 2

N. J. A. Sloane, Aug 31 2017

nonn

			For n=9, 2^n-1 = 511 with iterates 511->591->791->911, and 911 is the first prime, so a(7)=911.

Lars Blomberg, Table of n, a(n) for n = 2..270

Cf. A039654, A039655, A291301, A291302, A291777.

P(x) = {for(c=0,10^6,if(isprime(x),return(x),x=sigma(x)-1));-1}
vector(200,n,P(2^(n+1)-1)) \\ Lars Blomberg, Sep 01 2017

Added a(7) and a(13)-a(33) from Lars Blomberg, Sep 01 2017

A291777 a(n) = number of steps to reach a prime when x -> sigma(x)-1 is repeatedly applied to 2^n-1, or -1 if no prime is ever reached.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 3, 2, 9, 2, 0, 7, 3, 4, 0, 2, 0, 1, 4, 1, 4, 2, 3, 4, 2, 12, 22, 8, 0, 3, 3, 4, 3, 1, 2, 2, 3, 3, 4, 3, 13, 2, 16, 3, 8, 3, 14, 17, 9, 37, 4, 7, 4, 7, 11, 4, 3, 14, 0, 14, 8, 1, 6, 8, 73, 26, 10, 1, 32, 6, 10, 2, 6, 2, 33, 2, 4, 52, 12, 16

Offset: 2

N. J. A. Sloane, Aug 31 2017

nonn

			For n=9, 2^n-1 = 511 with iterates 511->591->791->911, and 911 is the first prime, so a(7)=3.

Lars Blomberg, Table of n, a(n) for n = 2..270

Cf. A039654, A039655, A291301, A291302, A291776.

C(x) = {for(c=0,10^5,if(isprime(x),return(c),x=sigma(x)-1));-1}
vector(200,n,C(2^(n+1)-1)) \\ Lars Blomberg, Sep 01 2017

a(13)-a(82) from Lars Blomberg, Sep 01 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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A291301 a(n) = prime that is eventually reached when x -> sigma(x)-1 is repeatedly applied to the product of the first n primes, or -1 if no prime is ever reached.

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A291776 a(n) = prime that is eventually reached when x -> sigma(x)-1 is repeatedly applied to 2^n-1, or -1 if no prime is ever reached.

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A291777 a(n) = number of steps to reach a prime when x -> sigma(x)-1 is repeatedly applied to 2^n-1, or -1 if no prime is ever reached.

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PARI

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