The home prime of a number is found by repeated concatenation
of its prime factors until a prime number is obtained. For example, the home prime of 9 is 311 since 9 = 3*3 giving 33 which is composite; 33 = 3*11 giving 311 which is a prime number. So, the home prime of n for n = 2, 3, 4 … are 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 733, 11, 233 … (Sequence A037274 of OEIS). The number of steps needed to obtain the home prime of n for n = 2, 3, 4 … are 0, 0, 2, 0, 1, 0, 13, 2, 4, 0, 1, 0, 5, 4, 4, 0, 1 … (Sequence A037273 of OEIS). It is believed that every number has a homeprime. The smallest number whose homeprime is not known is 49.
It is checked upto 109-th step and all resultant concatenations are composite. Similarly, the first few numbers whose homeprime is not known are 77, 49, 146, 246, 312, 320 … At least 50 concatenations of all these numbers are verified.