December 30, 2008

Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are called amicable numbers.

For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71 and 142 so d(284) = 220.

Evaluate the sum of all the amicable numbers under 10000.

(defun divisor-p (num divisor)
"Tests if divisor is a divisor of num"
(zerop (mod num divisor)))

(defun divisors (num)
"Returns a list of num's divisors"
(do ((i 2 (1+ i))
(result '(1)))
((> i (isqrt num)) (remove-duplicates result))
(when (divisor-p num i)
(setf result (append result (list i (/ num i)))))))

(defun sum-divisors (num)
"Sums the divisors of num"
(reduce #'+ (divisors num)))

(defun amicable-p (num)
"Tests if num is amicable"
(let ((divisors-sum (sum-divisors num)))
(when (and (= num (sum-divisors divisors-sum))
(not (= num divisors-sum)))

(defun amicable-sum (limit)
"Returns the sum of all amicable numbers below limit"
(do ((i 1 (1+ i))
(result 0))
((>= i limit) result)
(when (amicable-p i)
(incf result i))))

(defun euler-21 ()
(amicable-sum 10000))