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SICP exercise 1.33
You can obtain an even more general version of accumulate (see exercise 1.32) by introducing the notion of a filter on the terms to be combined. That is, combine only those terms derived from values in the range that satisfy a specified condition. The resulting filtered-accumulate abstraction takes the same arguments as accumulate, together with an additional predicate of one argument that specifies the filter. Write filtered-accumulate as a procedure. Show how to express the following using filtered-accumulate:
a. the sum of the squares of the prime numbers in the interval a to b (assuming that you have a prime? predicate already written)
b. the product of all the positive integers less than n that are relatively prime to n (i.e., all positive integers i < n such that GCD(i, n) = 1).
Here's a filtered accumulate procedure that generates an iterative process:
(define (filtered-accumulate combiner null-value predicate term a next b) (define (iter a result) (if (> a b) result (iter (next a) (if (predicate a) (combiner (term a) result) result)))) (iter a null-value))
Here's the requested solution to part a. of the problem, including all necessary helper functions, all of which have been defined in previous exercises or in the text:
(define (square x) (* x x)) (define (smallest-divisor n) (find-divisor n 2)) (define (divides? a b) (= (remainder b a) 0)) (define (find-divisor n test-divisor) (cond ((> (square test-divisor) n) n) ((divides? test-divisor n) test-divisor) (else (find-divisor n (next test-divisor))))) (define (next n) (if (= n 2) 3 (+ n 2))) (define (prime? n) (= n (smallest-divisor n))) (define (inc n) (+ n 1)) (define (sum-of-squares-of-primes a b) (filtered-accumulate + 0 prime? square a inc b))
Here are some tests for intervals [1, 10] and [4, 11]. Their answers should be
1 + 22 + 32 + 52 + 72 = 1 + 4 + 9 + 25 + 49 = 88
52 + 72 + 112 = 25 + 49 + 121 = 195
(sum-of-squares-of-primes 1 10)
(sum-of-squares-of-primes 4 11)
Those answers are correct.
Here's the requested solution to part b., using the gcd procedure from exercise 1.20:
(define (identity n) n) (define (gcd a b) (if (= b 0) a (gcd b (remainder a b)))) (define (product-of-relative-primes n) (define (relatively-prime-to-n? a) (= (gcd a n) 1)) (filtered-accumulate * 1 relatively-prime-to-n? identity 2 inc (- n 1)))
And a couple of tests using n = 8, 4 and 9. The answers should be