PWC 171 › Odd Abundant Numbers

This post is part of a series on Mohammad Anwar’s excellent Weekly Challenge, where hackers submit solutions in Perl, Raku, or any other language, to two different challenges every week. (It’s a lot of fun, if you’re into that sort of thing.)

This week’s tasks include a simple number theory calculation, and a language feature. Here’s a look at task 1.

Odd Abundant Numbers

Abundant numbers are numbers where the sum of the proper divisors is greater than the number. The first odd abundant number is 945. 945’s proper divisors are 1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, and 315. (Recall that we exclude 945 itself as a proper divisor.)

The sum of those divisors is 975, so therefore 945 is an abundant number.

Equivalently, a number is abundant if the sum of all divisors is greater than twice the number. We’ll see both variations in the solutions below.

I’m going to take you through a few different approaches. Since I don’t like repeating myself, let’s get some foreshadowing for task #2 going, by using a first class function:

sub n_odd_abundant(&$) {
    my ($is_abundant, $N) = @_;

    my @r;
    for (my $n = 3; $N; $n += 2) {
        if ($is_abundant->($n)) {
            push @r, $n;
            $N--
        }
    }

    @r;
}

The above code takes in a code ref ($is_abundant) and a limit ($N). We loop over all odd numbers, pushing any that pass the $is_abundant check to our result.

Although the above version suited my purposes better, it’s also possible to do this with an iterator, to avoid having to store the intermediate result:

sub odd_abundant_iterator(&) {
    my $is_abundant = shift;
    my $n = 1;
    
    sub {
        do { $n += 2 } until $is_abundant->($n);

        $n;
    }
}

Now that we have a framework for gathering abundant numbers, let’s try it a few different ways.

Brute force

The first way you might think to try is simply to brute force your way through every divisor of every number. This is O(n) for each number, and takes over a second to find the first 20 numbers:

sub n_abundant_naive {
    n_odd_abundant {
        my $n = shift;
        $n < sum grep { $n % $_ == 0 } 1..$n/2;
    } $_[0];
}

Using sqrt

Stopping at \(\sqrt{n} \) perhaps unsurprisingly brings the asymptotic time down to O(\(\sqrt{n} \)). Since divisors come in pairs, we can simply calculate the other divisor and avoid looping through most of the numbers:

sub n_abundant_sqrt {
    n_odd_abundant {
        my $n    = shift;
        my $sqrt = sqrt($n);

        my $sum  = sum map { $_,  $n / $_ } 
                      grep { $n % $_ == 0 } 1..$sqrt;

        $sum -= $sqrt if $sqrt == int $sqrt;
        
        2*$n < $sum;
    } $_[0];
}

It might not seem like a huge change, but the above code runs about 27 times faster than the naïve version, when asked to find the first 20 odd abundant numbers.

Using Math::Prime::Util divisor_sum

Just for fun, our old friend, Math::Prime::Util has a function that seems perfect for our needs: divisor_sum. It does what it says on the tin: it calculates the sum of the divisors of whatever number we give it.

sub n_abundant_mpu {
    n_odd_abundant {
        my $n = shift;
        my $sum = divisor_sum($n);

        2*$n < $sum;
    } $_[0];
}

This one is another 12 times faster than the sqrt solution, and 356 times faster than the naïve method. Great, right? Well, not so fast. Under the hood, the divisor_sum function is still finding all divisors for every number, so we’re still at \(O(n \sqrt{n}) \) time. It’s only faster because of the very tightly optimized C code under the hood.

Sieve

We can still do quite a bit better by realizing that since we’re looking for a whole bunch of abundant numbers, we’re repeating the same calculations over and over again, for every multiple of a number we’ve seen already. So as long as we’re willing to tweak the requirement slightly to find all abundant numbers below a given limit (although we’ll see how we can still accommodate the old calling syntax if we really want to), we can do much better:

sub n_abundant_sieve {
    my $lim = shift;
    my @r;

    my @div_sum; # Sum of divisors for each number
    for my $n (1..$lim) {
        $div_sum[$n*$_] += $n for 1..$lim/$n+1;
        push @r, $n if $n % 2 and 2*$n <= $div_sum[$n];
    }

    @r;
}

When called as n_odd_abundant_sieve(10000), this returns the first 23 odd abundant numbers, about twice as fast as the sqrt version returns 20. That’s because this algorithm runs in \(O(n \log{n}) \) time, which is strictly less than \(O(n \sqrt{n}) \) time for all \(n > 0 \). It grows significantly slower.

Sneaking up on the limit

In number theory circles, finding all numbers below a limit is usually just fine, but if we wanted to be a stickler for the parameters of the challenge and return only the first 20, we could simply call n_odd_abundant_sieve multiple times, doubling the limit every time, until we have at least 20 results. I wouldn’t bother, though.


You might be wondering, “how does this perform compared to the MPU version?”

For small values (the first 23 numbers in the sequence), the MPU version is faster by about 200%. However, when given more realistic limits, our better algorithm pulls ahead. It breaks even at about 100 numbers on my machine. By the time we ask for 150 numbers, the sieve is already 60% faster.

Benchmarks are an incredibly useful—and essential—tool when you need to write high performance code. Knowing that one algorithm is faster than another doesn’t always translate to real code. When dealing with implementation details, those hidden constants can really make a difference. What’s also really important, however, is knowing your requirements. If we’re only ever going to need to find less than 100 abundant primes, the MPU version might be a worthy choice. However, if we want to push it beyond that, then the better algorithm wins.

Of course, translating the sieve algorithm to tight C code would blow both of them out of the water at any value of n, since the sieve version grows strictly slower than the MPU version.

Raku

Here’s a quick port to Raku. If you’re only concerned with outputting the results, there’s no point in storing them first:

sub MAIN(Int $lim = 10000) {
    my @div_sum; # Sum of divisors for each number

    for 1...$lim -> $n {
        @div_sum[$n*$_] += $n for 1..$lim/$n+1;
        $n.say if $n % 2 and 2*$n <= @div_sum[$n];
    }

}

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