Project Euler Solution #23: Non-abundant sums

A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.

A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.

As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

Solution

With Problem 21, we figured out on how to calculate the sum of proper divisors already. We can recycle this knowledge here, since we need to decide based on this sum, if a number is a perfect number, an abundant or a deficient number. Since they already spoilered the upper limit of 28123, we need primes only up to \(\sqrt{28123}\) and can copy the sigma function over (I know, I should solve the prime generation with a sieve...):

function sigma(n) {
  var sum = 1;
  var primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167];

  if (n < 4) {
    return 1;
  }

  for (var i = 0; i < primes.length; i++) {

    var p = primes[i];

    if (0 === n % p) {

      var t = p * p;
      n /= p;
      while (0 === n % p) {
        t *= p;
        n /= p;
      }
      sum = sum * (t - 1) / (p - 1);
    }
    if (p * p > n) {
      break;
    }
  }

  if (n > 1) {
    sum *= n + 1;
  }
  return sum;
}

We know that \(\sigma\) is the sum of all divisors, not of all proper divisors. As such, abundant numbers can be checked with the following function, using the same caching mechanism as in Problem 21:

var cache = {};
function isAbundant(n) {

  if (n < 10)
    return false;

  if (cache[n]) {
    return cache[n];
  }
  return cache[n] = (sigma(n) - n > n);
}

With the isAbundant function, we can formulate a new function, which checks if a number is a sum of two abundant numbers:

function isSumOfTwoAbundants(n) {

  for (var i = 1; i <= n; i++) {
    if (isAbundant(i) && isAbundant(n - i))
      return true;
  }
  return false;
}

Since this function scans all numbers for abundant numbers over and over again, we can save all abundant numbers once, and reduce the search scope quite a bit:

var abundants = [];
for (var i = 1; i <= 28123; i++) {
  if (isAbundant(i))
    abundants.push(i);
}
function isSumOfTwoAbundants(n) {
  for (var i = 0; i < abundants.length; i++) {
    if (isAbundant(n - abundants[i]))
      return true;
  }
  return false;
}

However, we know the abundant numbers already but still go back to the isAbundant function. A much faster way is just summing all abundant numbers and check if they exceed our limit, which removes the isSumOfTwoAbundants function and makes it a lookup array:

var isSumOfTwoAbundants = new Array(28123 + 1);
for (var i = 0; i < abundants.length; i++) {
  for (var j = i; j < abundants.length; j++) {
    if (abundants[i] + abundants[j] <= 28123) {
      isSumOfTwoAbundants[abundants[i] + abundants[j]] = true;
    } else {
      break;
    }
  }
}

Which finally lets us compute the solution:

function solution() {

  var sum = 0;
  for (var i = 1; i <= 28123; i++) {
    if (!isSumOfTwoAbundants[i]) {
      sum+= i;
    }
  }
  return sum;
}