## 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)))      num)))(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))`