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 Oct. 2026, the sequence has reached a 257-digit
composite number after 118 iterations,
counting the initial 49 as iteration 0. It has a 251-digit
composite cofactor whose factorization
remains incomplete (De Geest 2022, FactorDB).
As of Oct. 2026, there are 28 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,
FactorDB).
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