%I #148 Apr 11 2024 11:19:02

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

%N 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.

%C 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

%C 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

%C The number n = 3^6 * 2331961591220850480109739369 * 21313644799483579440006455257 is a near-miss 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

%C 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

%C 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

%C 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

%C The value of a(20) is presently unknown (see A195265).

%H R. J. Mathar, <a href="/A195264/b195264.txt">Table of n, a(n) for n = 1..19</a>

%H J. H. Conway, <a href="/A195264/a195264_4.pdf">Five $1000 Problems</a>, Oct 10 2014 (recorded by Bill Cheswick).

%H Alonso del Arte and Sean A. Irvine, <a href="/A195264/a195264_2.txt">Table of n, a(n) for n = 1..1000</a>

%H Hans Havermann, <a href="http://chesswanks.com/seq/a195264/">Table of n, a(n) for n = 1..10000</a> (includes links to lengthy (>40) and unknown-outcome evolutions, and a list of unfactored composites in the unknowns' last step)

%H Hans Havermann, <a href="/A195264/a195264_2.pdf">Table of n, a(n) for n = 1..10000</a> (includes links to lengthy (>40) and unknown-outcome evolutions, and a list of unfactored composites in the unknowns' last step) [Cached copy, pdf version as of May 08 2018, with permission]

%H Hans Havermann, <a href="http://gladhoboexpress.blogspot.ca/2017/06/13532385396179-precursors.html">13532385396179 precursors</a>

%H Hans Havermann, <a href="/A195264/a195264_1.pdf">13532385396179 precursors</a> [Cached copy, pdf version as of May 08 2018, with permission]

%H OEIS50 DIMACS Conference on Challenges of Identifying Integer Sequences, <a href="https://vimeo.com/109815595">Problem Session 2, Oct 10 2014</a>, J. H. Conway, Five $1000 Problems (starting at about 06.44). This sequence is mentioned in the fifth problem, starting at around 19:30.

%H Tony Padilla and Brady Haran, <a href="https://www.youtube.com/watch?v=3IMAUm2WY70">13532385396179</a>, Numberphile video, 2017

%H N. J. A. Sloane, <a href="/A195264/a195264.pdf">Confessions of a Sequence Addict (AofA2017)</a>, slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.

%H N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, <a href="https://vimeo.com/237029685">Part I</a>, <a href="https://vimeo.com/237030304">Part 2</a>, <a href="https://oeis.org/A290447/a290447_slides.pdf">Slides.</a> (Mentions this sequence)

%H Doron Zeilberger, <a href="http://sites.math.rutgers.edu/~zeilberg/oeis50.html">Videos of Talks Delivered in SLOANE75/OEIS50 DIMACS Conference on Challenges of Identifying Integer Sequences</a> (see Problem Session 2, Oct 10 2014)

%e 4 = 2^2 -> 22 =2*11 -> 211, prime, so a(4) = 211.

%e 9 = 3^2 -> 32 = 2^5 -> 25 = 5^2 -> 52 = 2^2*13 -> 2213, prime, so a(9)=2213.

%t 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 *)

%t fn[n_] := FromDigits[Flatten[IntegerDigits[DeleteCases[Flatten[

%t FactorInteger[n]], 1]]]];

%t Table[NestWhile[fn, n, # != 1 && ! PrimeQ[#] &], {n, 19}] (* _Robert Price_, Mar 15 2020 *)

%o (PARI) a(n)={n>1 && while(!ispseudoprime(n), n=A080670(n));n} \\ _M. F. Hasler_, Oct 12 2014

%Y A variant of the home primes, A037271. Cf. A080670, A195265 (trajectory of 20), A195266 (trajectory of 105), A230305, A084318. A230627 (base-2), A290329 (base-3)

%K nonn,base,more

%O 1,2

%A _N. J. A. Sloane_, Sep 14 2011, based on discussions on the Sequence Fans Mailing List by _Alonso del Arte_, _Franklin T. Adams-Watters_, _D. S. McNeil_, _Charles R Greathouse IV_, _Sean A. Irvine_, and others