; mertens' conjecture
(define range
(case-lambda
((stop) (range 0 stop (if (negative? stop) -1 1)))
((start stop) (range start stop (if (< start stop) 1 -1)))
((start stop step)
(let ((le? (if (negative? step) >= <=)))
(let loop ((x start) (xs (list)))
(if (le? stop x) (reverse xs)
(loop (+ x step) (cons x xs))))))
(else (error 'range "too many arguments"))))
(define (factors n)
(let ((wheel (vector 1 2 2 4 2 4 2 4 6 2 6)))
(let loop ((n n) (f 2) (w 0) (fs (list)))
(if (< n (* f f)) (reverse (cons n fs))
(if (zero? (modulo n f))
(loop (/ n f) f w (cons f fs))
(loop n (+ f (vector-ref wheel w))
(if (= w 10) 3 (+ w 1)) fs))))))
(define (moebius n)
(if (< n 1) (error 'moebius "must be positive")
(if (= n 1) 1
(let loop ((m 1) (f 0) (fs (factors n)))
(if (null? fs) m
(if (= f (car fs)) 0
(loop (- m) (car fs) (cdr fs))))))))
(define (mertens n)
(do ((k 1 (+ k 1))
(m 0 (+ m (moebius k))))
((< n k) m)))
(define (a008683 n) ; moebius function
(map moebius (range 1 (+ n 1))))
(display (a008683 25)) (newline)
(define (a002321 n) ; mertens function
(map mertens (range 1 (+ n 1))))
(display (a002321 20)) (newline)
(define (a028442 n)
; numbers k such that mertens(k) == 0
(let loop ((k 1) (M 0) (ks (list)))
(if (< n k) (reverse ks)
(let* ((m (moebius k)) (M (+ M m)))
(if (zero? M)
(loop (+ k 1) M (cons k ks))
(loop (+ k 1) M ks))))))
(display (a028442 200)) (newline)
(define (a100306 n)
; numbers k such that moebius(k) == mertens(k)
(let loop ((k 1) (M 0) (ks (list)))
(if (< n k) (reverse ks)
(let* ((m (moebius k)) (M (+ M m)))
(if (= m M)
(loop (+ k 1) M (cons k ks))
(loop (+ k 1) M ks))))))
(display (a100306 200)) (newline)