The prime 
reached starting from a number 
, concatenating its prime factors, and repeating until a prime
is reached. For example, for 
,
 |
so 311 is the home prime of 9. For 
, 3, ..., the first few are 2, 3, 211, 5, 23, 7, 3331113965338635107,
311, 773, ... (OEIS A037274). Probabilistic
arguments give exactly zero for the chance that the sequence of integers starting
at a given number 
contains no prime, so a home prime should exist for every positive integer.
Since prime numbers have trivial home primes (themselves), attention can be restricted to composite numbers. The numbers of steps to arrive at a home prime for composite numbers 4, 6, 8, 9, ... are 2, 1, 13, 2, 4, 1, 5, 4, 4, 1, 15, 1, ... (OEIS A037271), and the primes they reach are 211, 23, 3331113965338635107, 311, 773, 223, ... (OEIS A037272).
The largest home prime for 
is 
, although its value is
not known. The first few terms in the home prime sequence for 49 are 49, 77, 711,
3379, 31109, 132393, 344131, ... (OEIS A056938).
As of April 2011, computation of the sequence for this number is currently stalled
at the 109th step on a 232-digit number which has yet to be completely factored.
As of April 2011, there are 30 unknown 
with 
(excluding values of 
such as 77 which occur in the home prime sequence of a smaller
number), the first few of which are 49, 146, 242, 312, 320, ... (Bonath).
."