

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; or 1 if a prime is never reached.


8



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

It seems likely that every number does eventually reach a prime, as for home primes (A037271). However, since it is possible to have A080670(n) < n, it is conceivable that some starting values reach a nontrivial loop.
J. H. Conway offers $1000 for a proof that every number eventually reaches a prime  see link.  N. J. A. Sloane, Oct 15 2014
This 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


LINKS

R. J. Mathar, Table of n, a(n) for n = 1..19
J. H. Conway, Five $1000 Problems (This sequence is mentioned in the fifth problem)
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)


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 *)


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.
Sequence in context: A231387 A160759 A191835 * A037274 A037275 A142960
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



