PWC171 - Abundant Number


Here we are with TASK #1 from The Weekly Challenge #171. Enjoy!

The challenge

Write a script to generate first 20 Abundant Odd Numbers.

According to wikipedia,

A number n for which the sum of divisors σ(n) > 2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n) > n.

For example, 945 is the first Abundant Odd Number.

Sum of divisors:
1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975

The questions

I’m always tempted to ask about what first means in these challenges. In these number theory stuff, it usually means least non-negative ones, but it’s always stated as first and this probably leaves space to… is it good to read it as the first ones that I can find?.

The solution

The question is not pure nitpicking this time. As the wikipedia page explains, every (integer) multiple of an abundant number is itself an abundant number, so multiplying 945 times 1, 3, …, 39 (odds numbers between 0 and 40) will bot give us an odd number and an abundant number. So easy to calculate and so light on the environment.

Alas, let’s take the parsimonious path and find the lowest ones instead, starting with Raku:

#!/usr/bin/env raku
use v6;
sub MAIN (Int:D $n where * > 0 = 20) {
   my @abundants;
   my $candidate = 945;
   while @abundants < $n {
      @abundants.push: $candidate if is-abundant($candidate);
      $candidate += 2;
   put @abundants.join(', ');

sub is-abundant (Int:D $n) { $n < [+] proper-positive-divisors($n) }

sub proper-positive-divisors (Int:D $n is copy where * != 0) {
   $n = $n.abs;
   my (@lows, @highs) = 1,;
   my ($lo, $hi) = (2, $n);
   while $lo < $hi {
      if $n %% $lo {
         @lows.push: $lo;
         $hi = ($n / $lo).Int;
         @highs.unshift: $hi if $hi != $lo;
   return [@lows.Slip, @highs.Slip];

It’s pure composition of bricks. Sub proper-positive-divisors takes care to give us all… proper, positive divisors. This is used inside the check function is-abundant to test the condition. An outer loop (the while in the MAIN sub) takes care of finding out the lowest EHR first 20.

The function for finding divisors is marginally interesting. As soon as we find a divisor, we also find its complementary in the product to get to the original number, so we can accumulate divisors from the bottom and from the top. At this point, there’s no reason to go beyond when the lower candidate gets past its complementary.

It’s interesting that this approach might not give us any edge at all with respect to iterating more and not caring about the complementaries. This is because - more or less! - there is a division check with the %% operator and a real division. Assuming the two take the same resources, we might just as well ignore the complementary ang move on.

The Perl counterpart is a faithful translation:

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

my $n = shift // 20;
my @abundants;
my $candidate = 945;
while (@abundants < $n) {
   push @abundants, $candidate if is_abundant($candidate);
   $candidate += 2;
say join ', ', @abundants;

sub is_abundant ($n) { $n < sum(proper_positive_divisors($n)) }

sub proper_positive_divisors ($n) {
   return unless $n;
   $n = -$n if $n < 0;
   my (@lows, @highs) = (1);
   my ($lo, $hi) = (2, $n);
   while ($lo < $hi) {
      if ($n % $lo == 0) {
         push @lows, $lo;
         $hi = int($n / $lo);
         unshift @highs, $hi if $hi != $lo;
   return (@lows, @highs);

It feels less sophisticated but heck does this feel natural!

Stay safe folks!

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