

A195264


Iterate x > A080670(x) (replace x with the concatenation of the primes and exponents in its prime factorization) starting at n until reach 1 or a prime (which is then the value of a(n)); or a(n) = 1 if a prime is never reached.


27



1, 2, 3, 211, 5, 23, 7, 23, 2213, 2213, 11, 223, 13, 311, 1129, 233, 17, 17137, 19
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OFFSET

1,2


COMMENTS

J. H. Conway offered $1000 for a proof or disproof for his conjecture that every number eventually reaches a 1 or a prime  see OEIS50 link.  N. J. A. Sloane, Oct 15 2014
However, James Davis has discovered that a(13532385396179) = 1. This number D = 13532385396179 = (1407*10^5+1)*96179 = 13*53^2*3853*96179 is clearly fixed by the map x > A080670(x), and so never reaches 1 or a prime.  Hans Havermann, Jun 05 2017
The number n = 3^6 * 2331961591220850480109739369 * 21313644799483579440006455257 is a nearmiss for another nonprime fixed point. Unfortunately here the last two factors only look like primes (they have no prime divisors < 10), but in fact both are composite.  Robert Gerbicz, Jun 07 2017
The number D' = 13^532385396179 maps to D and so is a much larger number with a(D') = 1. Repeating this process (by finding a prime prefix of D') should lead to an infinite sequence of counterexamples to Conway's conjecture.  Hans Havermann, Jun 09 2017
The first 47 digits of D' form a prime P = 68971066936841703995076128866117893410448319579, so if Q denotes the remaining digits of 13^532385396179 then D'' = P^Q is another counterexample.  Robert Gerbicz, Jun 10 2017
This sequence is different from A037274. Here 8 = 2^3 > 23 (a prime), whereas in A037274 8 = 2^3 > 222 > ... > 3331113965338635107 (a prime).  N. J. A. Sloane, Oct 12 2014
The value of a(20) is presently unknown (see A195265).


LINKS

R. J. Mathar, Table of n, a(n) for n = 1..19
Alonso del Arte and Sean A. Irvine, Table of n, a(n) for n = 1..1000
Hans Havermann, Table of n, a(n) for n = 1..10000 (includes links to lengthy (>40) and unknownoutcome evolutions, and a list of unfactored composites in the unknowns' last step)
Hans Havermann, Table of n, a(n) for n = 1..10000 (includes links to lengthy (>40) and unknownoutcome evolutions, and a list of unfactored composites in the unknowns' last step) [Cached copy, pdf version as of May 08 2018, with permission]
Hans Havermann, 13532385396179 precursors
Hans Havermann, 13532385396179 precursors [Cached copy, pdf version as of May 08 2018, with permission]
OEIS50 DIMACS Conference on Challenges of Identifying Integer Sequences, Problem Session 2, Oct 10 2014, J. H. Conway, Five $1000 Problems (starting at about 06.44). This sequence is mentioned in the fifth problem, starting at around 19:30.
Tony Padilla and Brady Haran, 13532385396179, Numberphile video, 2017
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
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)
Doron Zeilberger, Videos of Talks Delivered in SLOANE75/OEIS50 DIMACS Conference on Challenges of Identifying Integer Sequences (see Problem Session 2, Oct 10 2014)


EXAMPLE

4 = 2^2 > 22 =2*11 > 211, prime, so a(4) = 211.
9 = 3^2 > 32 = 2^5 > 25 = 5^2 > 52 = 2^2*13 > 2213, prime, so a(9)=2213.


MATHEMATICA

f[1] := 1; f[n_] := Block[{p = Flatten[FactorInteger[n]]}, k = Length[p]; While[k > 0, If[p[[k]] == 1, p = Delete[p, k]]; k]; FromDigits[Flatten[IntegerDigits[p]]]]; Table[FixedPoint[f, n], {n, 19}] (* Alonso del Arte, based on the program for A080670, Sep 14 2011 *)
fn[n_] := FromDigits[Flatten[IntegerDigits[DeleteCases[Flatten[
FactorInteger[n]], 1]]]];
Table[NestWhile[fn, n, # != 1 && ! PrimeQ[#] &], {n, 19}] (* Robert Price, Mar 15 2020 *)


PROG

(PARI) a(n)={n>1 && while(!ispseudoprime(n), n=A080670(n)); n} \\ M. F. Hasler, Oct 12 2014


CROSSREFS

A variant of the home primes, A037271. Cf. A080670, A195265 (trajectory of 20), A195266 (trajectory of 105), A230305, A084318. A230627 (base2), A290329 (base3)
Sequence in context: A160759 A191835 A317550 * A329181 A316941 A037274
Adjacent sequences: A195261 A195262 A195263 * A195265 A195266 A195267


KEYWORD

nonn,base,more


AUTHOR

N. J. A. Sloane, Sep 14 2011, based on discussions on the Sequence Fans Mailing List by Alonso del Arte, Franklin T. AdamsWatters, D. S. McNeil, Charles R Greathouse IV, Sean A. Irvine, and others


STATUS

approved



