PWC168 - Home Prime


On with TASK #2 from The Weekly Challenge #168. Enjoy!

The challenge

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.


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.

The questions

What’s with all these primes? I tell you, our fine host is on to something!

The solution

This week I decided to go for the unusual path and somehow let Perl show off its muscles. So we’re going fully functional with this otherwise barely readable solution.

#!/usr/bin/env perl
use v5.24;
use warnings;
use experimental 'signatures';
no warnings 'experimental::signatures';

use ntheory qw< factor >;

my $n = shift // 10;
say home_prime($n);

sub home_prime {
   my $next = join '', factor($_[0]);
   return $next if $next eq $_[0];
   $_[0] = $next;
   goto &home_prime;

By the examples themselves, we know that a recursive function must work fine. BUT we don’t want to clutter our call stack, right? We need tail call optimization, right? Well, I don’t know if Perl has it, in any version, but I do know how to simulate it:

  • arrange @_ properly
  • use goto (in one of its good incarnations inside the language).

So there you go, functional loops!

Kudos to DANAJ for saving our coordinate axes with the excpetional ntheory.

ON THE OTHER HAND, Raku might one day have tail call optimization, but we’re not there yet apparently. So we’ll go… fully iterative this time, thanks to a loop function that expresses perfectly the fact that we might go on and on for ages:

#!/usr/bin/env raku
use v6;
sub MAIN (Int:D $n where * > 1 = 10) { put home-prime($n) }

sub home-prime ($n is copy) {
   loop {
      my $m = factors($n).join('').Int;
      return $n if $n == $m;
      $n = $m;

sub factors (Int $remainder is copy) {
   return 1 if $remainder <= 1;
   state @primes = 2, 3, 5, -> $n is copy {
      repeat { $n += 2 } until $n %% none @primes ... { $_ * $_ >= $n }
   } ... *;
   gather for @primes -> $factor {
      if $factor * $factor > $remainder {
         take $remainder if $remainder > 1;

      # How many times can we divide by this prime?
      while $remainder %% $factor {
         take $factor;
         last if ($remainder div= $factor) === 1;

The factors function is by none other than Tim Toady himself! What a honor to linger on the shoulders of the giants!

Well, thanks everybody for making this challenge such effortless and a pleasure to code!

Stay safe!

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