A239448 Limiting value of the iterated process of factoring n and concatenating prime powers (in decimal) in the order of increasing primes.

1, 2, 3, 4, 5, 23, 7, 8, 9, 25, 11, 43, 13, 27, 1129, 16, 17, 29, 19, 36389, 37, 211, 23, 83, 25, 3251, 27, 47, 29, 547, 31, 32, 311, 31397, 1129, 49, 37, 373, 313, 3137, 41, 379, 43, 3137, 36389, 223, 47, 163, 49, 71443, 317, 31123, 53, 227, 773, 983, 1129, 229, 59, 3529, 61, 31237, 97, 64, 2719

Offset: 1

Author

James G. Merickel, Apr 27 2014

Keywords

• nonn
• base

Comments

This is the number reached by iterating the process used in generating A080695, and is similar to the sequence of home primes (A037274), differing by concatenation not of individual primes but, rather, of prime powers. The author suggests using the term 'away number' in a way analogous to 'home prime', thinking that merger of capital 'H' with the symbol '^' suggests capital 'A', etc. (and prime powers--not merely primes--can result, depending on the number). Each positive integral value n should, with heuristic probability 1, have such an away number, and 303 is the first number presenting any challenge finding such (at time of submission). a(1)=1 as a convention, and the numbers that are their own away numbers are the members of A000961.

Examples

			A080695(15)=35, A080695(35)=57, A080695(57)=319, A080695(319)=1129, and A080695(1129)=1129. So, a(15)=1129.
		

Crossrefs

Cf. A080695, A037274, A000961.

Programs

Mathematica

f[n_]:=Module[{l=FactorInteger[n]},
Do[l[[i]]=l[[i,1]]^l[[i,2]],{i,1,Length[l]}];
l=FromDigits[Flatten[IntegerDigits/@l]]];
fp[n_]:=FixedPoint[f,n];fp/@Range[65] (* Ivan N. Ianakiev, Aug 02 2015 *)

PARI

{
print1(1", ");n=2;
while(1,
  N=n;f=factor(N);m=matsize(f)[1];
  while(m!=1,
    N=f[1,1]^f[1,2];
    for(i=2,m,
      e=10;k=f[i,1]^f[i,2];
      while(k>e,e*=10);N*=e;N+=k);
    f=factor(N);m=matsize(f)[1]);
  print1(N", ");n++)
}

Extensions

a(65) corrected by Ivan N. Ianakiev, Aug 02 2015