PWC170 - Primorial Numbers


Here we are with TASK #1 from The Weekly Challenge #170. Enjoy!

The challenge

Write a script to generate first 10 Primorial Numbers.

Primorial numbers are those formed by multiplying successive prime numbers.

For example,

P(0) = 1    (1)
P(1) = 2    (1x2)
P(2) = 6    (1x2×3)
P(3) = 30   (1x2×3×5)
P(4) = 210  (1x2×3×5×7)

The questions

I would only ask what does it mean to generate the first 10 of the lot. Is P(0) to be considered the first one, so that we should stop at P(9)? Or is P(1) the real first primorial number, because it’s the one involving the first prime?

Well, our fine host took the effort to put P(0) in the example, so I’ll assume it’s also the first one.

The solution

I knew that there was some sleight of hand to do this very compactly in Raku, and I found one way. I’m curious to read more solution in the days to come.

Let’s go step-wise. I know how to generate a (lazy) infinite list of positive integers:

1 .. *

and this can be filtered for 1 or primes, keeping the laziness intact:

(1 .. *).grep({$_ == 1 || .is-prime})

Now, though, we have to do the products. If we were to calculate any single primorial, we might first isolate the terms of interest with some slicing:

# $n leads to the $n-th primorial P($n - 1)
(1 .. *).grep({$_ == 1 || .is-prime})[^$n]

Then we might apply the hyperoperator [*] to this slice and get our primorial:

my $nth-primorial = [*] (1 .. *).grep({$_ == 1 || .is-prime})[^$n]

Alas, we have to produce a new sequence here…

Sorry, I meant that we have to produce a new sequence here:

[produce] is similar to reduce, but returns a list with the accumulated values instead of a single result.

So we first produce, then we slice to the amount of items that we need:

#!/usr/bin/env raku
use v6;
sub MAIN (Int:D $n where * > 0 = 10) {
   .put for (1 .. *).grep({$_ == 1 || .is-prime}).produce(&[*]).[^$n];

Let’s get to Perl now. There’s none of this laziness craziness, hyperstuff or so, but we have our old friends iterators and some basic golfing capabilities to make the readers scratch their heads a bit:

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

use ntheory 'next_prime';

my $it = primorial_it();
say $it->() for 1 .. shift || 10;

sub primorial_it ($n = 1, $p = 1) {
   sub { (($p, $n) = ($p * $n, next_prime($n)))[0] };

In the iterator sub we use two variables (which are defined as arguments, just to spare a line of code… sorry!), one is $p which keeps the ever-growing product, and one is $n which keeps track of the next prime to use.

The fun thing is that the list assignment first calculates the list on the right hand side, then it does the assignment. Hence, the assignment between the two lists actually means the same as:

$p = $p * $n;
$n = next_prime($n);

and returns ($p, $n). As we’re interested into returning $p, why not return the first element of this small list?

You know, this can be golfed a bit and gain in readability:

sub primorial_it ($n = 1, $p = 1) {
 # sub { (($p, $n) = ($p * $n, next_prime($n)))[0] };
   sub { ($p, $n) = ($p * $n, next_prime($n)); $p };

It’s now clear that we want to return $p, yay!

Last considerations:

  • ntheory just rocks, thanks DANAJ;
  • I know the product is going to become a non-integer quite soon and I would need to use big integers… but we’re requested to cope with the first 10 items, and the stock integers are fine for this.

Stay safe!

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