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A195264 Iterate x -> A080670(x) (replace x by 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 (list; graph; refs; listen; history; text; internal format)
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 unknown-outcome 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. Adams-Watters, D. S. McNeil, Charles R Greathouse IV, Sean A. Irvine, and others.

STATUS

approved

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Last modified March 22 22:05 EDT 2017. Contains 283901 sequences.