023.js

// 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.

var isAbundant = function(){
    var memo = [];
    return function(n) {
        var is = memo[n];
        if (typeof is !== 'boolean') {
            var sum = 0;
            for (var i = 1; i <= n/2; i++) {
                if (n % i === 0) {
                        sum += i;
                }
            }
            is = sum > n;
            memo[n] = is;
        }
        return is;
    };
}();

function isSumOfTwoAbundants(n) {
    for (var i = 1; i <= n/2; i++) {
        if (isAbundant(i) && isAbundant(n-i)) {
            return true;
        }
    }
    return false;
}

function sumNonAbundant(n) {
    var sum = 0;
    for (var i = 1; i <= n; i++) {
        if (isSumOfTwoAbundants(i)) continue;
        sum += i;
    }
    return sum;
}

console.log( sumNonAbundant(28123) );