The home prime 
of an integer 
, when it exists, is the prime
number reached by repeatedly concatenating the decimal representations of the
prime factors in increasing order, including repeated
factors, until a prime number 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). It is conjectured
that this process reaches a prime number for every
integer 
; probabilistic heuristics do not prove termination.
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).
Whether the process starting from 49 reaches a prime number remains unknown. Its first step gives 77, so 
if these home primes exist. 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 was stalled at the 109th
step on a 232-digit number which had not yet been 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).
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