In number theory, the '''home prime HP(n)''' of an integer '''n''' greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions.

The traditional notation has the prefix "HP" and a postfix count of the number of iterations until the home prime is found (if the count is greater than 0), for instance HP4(2) === HP22(1) === 211 is the same as saying the home prime of 4 needs 2 iterations and is the same as the home prime of 22 which needs 1 iteration, and (both) resolve to 211, a prime.

Prime numbers are their own home prime;

So: 

    HP2 = 2
    
    HP7 = 7

If the integer obtained by concatenating increasing prime factors is not prime, iterate until you reach a prime number; the home prime.

    HP4(2) = HP22(1) = 211
    HP4(2) = 2 × 2 => 22; HP22(1) = 2 × 11 => 211; 211 is prime  
    
    HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773
    HP10(4) = 2 × 5 => 25; HP25(3) = 5 × 5 => 55; HP55(2) = 5 × 11 => 511; HP511(1) = 7 × 73 => 773; 773 is prime  


;Task

* Find and show here, on this page, the home prime iteration chains for the integers 2 through 20 inclusive.


;Stretch goal
    
* Find and show the iteration chain for 65.


;Impossible goal

* Show the the home prime for HP49.


;See also

;* [[oeis:A037274|OEIS:A037274 - Home primes for n >= 2]]
;* [[oeis:A056938|OEIS:A056938 - Concatenation chain for HP49]]

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