PWC156 - Pernicious Numbers


Here we are with TASK #1 from The Weekly Challenge #156. Enjoy!

The challenge

Write a script to permute first 10 Pernicious Numbers.

A pernicious number is a positive integer which has prime number of ones in its binary representation.

The first pernicious number is 3 since binary representation of 3 = (11) and 1 + 1 = 2, which is a prime.

Expected Output

3, 5, 6, 7, 9, 10, 11, 12, 13, 14

The questions

I’m not sure what we mean by permute the first 10 pernicious numbers, because the expected output only lists them in order. That’s probably a typo for compute.

The count to evaluate is precise, as it usually is, so no more questions. Were it not so specific, I’d at least ask for a ballpark.

The solution

There are so few pernicious numbers to calculate that it might be done by hand. The first three prime numbers are 2, 3, and 5, which respectively yield minimal pernicious numbers of 3 (11), 7 (111), and 31 (11111).

Before hitting 31, every number that has exactly either two or three bits is good for us, so e.g. 5 (101), 6 (110) and so on;

     0  N (0 "1")
     1  N (1 "1")
    10  N (1)
    11  Y (2 "1")
   100  N (1)
   101  Y (2)
   110  Y (2)
   111  Y (3 "1")
  1000  N (1)
  1001  Y (2)
  1010  Y (2)
  1011  Y (3)
  1100  Y (2)
  1101  Y (3)
  1110  Y (3)
  1111  N (4 "1")

This list alone already contains enough pernicious numbers to answer the challenge.

I’m usually tempted to generalize these challenges to account for an unspecified number of outputs, possibly in an efficient incremental way. This time, though, I decided to be very lazy. I mean very lazy. So it’s basically a matter of programming the most brutey thing I can come up with.

Raku allows us to be very brutey:

#!/usr/bin/env raku
use v6;

sub MAIN (Int:D $N = 10) {
   my @pernicious;
   my $k = 0;
   while @pernicious < $N {
      @pernicious.push: $k if is-pernicious($k);
   @pernicious.join(', ').put;

sub is-pernicious (Int:D $n where * >= 0) { $n.base(2) }

I mean, what could I ask more? Turning to base 2? Check. Counting the ones with a few keystrokes? Check. Testing for primality? Check. Checking the inputs? Check.

This probably does not scale exceptionally but who knows? And, more importantly, in this week I feel very much like who cares?

Perl needs gathering some batteries externally, but it’s OK anyway. Here we are really being very, very brutey, by summoning ntheory just to use is_prime to check primality of integers up to 3 (we will stop afterwards). This is what brute force is like.

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

use FindBin '$Bin';
use lib "$Bin/local/lib/perl5";
use ntheory 'is_prime';

my $N = shift // 10;
my @pernicious;
my $k = 0;
while (@pernicious < $N) {
   push @pernicious, $k if is_pernicious($k);
say join ', ', @pernicious;

sub is_pernicious ($n) {
   my $count = 0;
   while ($n) {
      ++$count if $n & 1;
      $n >>= 1;
   return is_prime($count);

Counting the bits that are set is done differently here, in a good old mask & shift fashion. At every round, we increment $count if the last bit is a 1, then shift the whole input number by one position to the right. Rinse and repeat until there’s no more bits left (i.e. $n drops to 0).

I’m still in awe to have called upon ntheory in this challenge. I’m proud and ashamed at the same time.

I encourage everyone to stay safe, even more so people who are suffering from the war. I really do hope this is going to end soon 😢 For everyone else, though, it seems that media is mostly focused on Ukraine these days, for good reasons, but please consider carefully your interactions and their consequences - I hear about a lot of people getting Covid-19.

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