ホーム - Project Euler

Problem 29¶

Distinct Powers¶

Consider all integer combinations of $a^b$ for $2 \le a \le 5$ and $2 \le b \le 5$:

\begin{matrix} 2^2=4, &2^3=8, &2^4=16, &2^5=32\\ 3^2=9, &3^3=27, &3^4=81, &3^5=243\\ 4^2=16, &4^3=64, &4^4=256, &4^5=1024\\ 5^2=25, &5^3=125, &5^4=625, &5^5=3125 \end{matrix} If they are then placed in numerical order, with any repeats removed, we get the following sequence of $15$ distinct terms: $$4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125.$$ How many distinct terms are in the sequence generated by $a^b$ for $2 \le a \le 100$ and $2 \le b \le 100$?

個別のべき乗¶

$2 \le a \le 5$ と $2 \le b \le 5$ について, $a^b$ を全て考えてみよう:

\begin{matrix} 2^2=4, &2^3=8, &2^4=16, &2^5=32\\ 3^2=9, &3^3=27, &3^4=81, &3^5=243\\ 4^2=16, &4^3=64, &4^4=256, &4^5=1024\\ 5^2=25, &5^3=125, &5^4=625, &5^5=3125 \end{matrix} これらを小さい順に並べ, 同じ数を除いたとすると, 15個の項を得る: $$4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125.$$ $2 \le a \le 100$, $2 \le b \le 100$ で同じことをしたときいくつの異なる項が存在するか?

(import (scheme base)
        (gauche base)
        (scheme sort))

(define (unique lis)
  (let loop ([prev #f] [rest lis] [result '()])
    (cond
      [(null? rest) (reverse result)]
      [(and prev
            (= prev (car rest)))
       (loop prev (cdr rest) result)]
      [else
        (loop (car rest)
              (cdr rest)
              (cons (car rest) result))])))

(define (powers a b)
  (assume (exact-integer? a))
  (assume (exact-integer? b))
  (assume (<= 2 a))
  (assume (<= 2 b))
  (fold (^[a lis]
          (append (fold (^[b lis]
                          (cons (expt a b) lis))
                        '()
                        (iota (- b 1) 2))
                  lis))
        '()
        (iota (- a 1) 2)))

(define answer-29
  (length (unique (list-sort < (powers 100 100)))))

(format #t "29: ~d~%" answer-29)