Home prime

You are given an integer greater than 1. Write a script to find the home prime of the given number.

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. Further information can be found on Wikipedia and OEIS.

Examples

Example 1:
As given in the Wikipedia page:
HP(10) = 773, as 10 factors as 2×5 yielding 
HP10(1) = 25, 25 factors as 5×5 yielding 
HP10(2) = HP25(1) = 55, 55 = 5×11 implies 
HP10(3) = HP25(2) = HP55(1) = 511, and 511 = 7×73 gives 
HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773, 
a prime number.

Analysis

The home prime $HP(n) of an integer $n is obtained by concatenating its prime factors (in ascending order) to form a new number and then repeating this process until the result is prime. Perl is good at switching between the binary and character representation of a number as the context demands, so if we are allowed factor() and is_prime() from Math::Util this task is fairly trivial using recursion.

This works in microseconds up to the 48th home prime, but the 49th is apparently yet to be solved.

Peter's use of f(23) is quite interesting. The short and precise discussion makes it all fun to read. Keep it up great work.

#!/usr/bin/perl

# Peter Campbell Smith - 2022-06-06
# PWC 168 task 2

use v5.28;
use strict;
use warnings;
use utf8;
use Math::Prime::Util qw[factor is_prime];

my ($test, $hp);

# loop over 2 to 48 (49 is known to be difficult!)
for $test (2 .. 48) {
    $hp = home($test);
    say qq[hp($test) = $hp];
}

sub home {
    
    my (@prime_factors, $concat);
    
    # get prime factors
    @prime_factors = factor(shift);
    
    # concatenate them
    $concat .= $_ for @prime_factors;
    
    # either we have an answer or need to go round again
    return $concat if is_prime($concat);
    home($concat);
}

6 lines of code

Output from script

hp(2) = 2
hp(3) = 3
hp(4) = 211
hp(5) = 5
hp(6) = 23
hp(7) = 7
hp(8) = 3331113965338635107
hp(9) = 311
hp(10) = 773
hp(11) = 11
hp(12) = 223
hp(13) = 13
hp(14) = 13367
hp(15) = 1129
hp(16) = 31636373
hp(17) = 17
hp(18) = 233
hp(19) = 19
hp(20) = 3318308475676071413
hp(21) = 37
hp(22) = 211
hp(23) = 23
hp(24) = 331319
hp(25) = 773
hp(26) = 3251
hp(27) = 13367
hp(28) = 227
hp(29) = 29
hp(30) = 547
hp(31) = 31
hp(32) = 241271
hp(33) = 311
hp(34) = 31397
hp(35) = 1129
hp(36) = 71129
hp(37) = 37
hp(38) = 373
hp(39) = 313
hp(40) = 3314192745739
hp(41) = 41
hp(42) = 379
hp(43) = 43
hp(44) = 22815088913
hp(45) = 3411949
hp(46) = 223
hp(47) = 47
hp(48) = 6161791591356884791277