Project Euler

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# [[Problem 63]]

## Powerful Digit Counts

The $5$-digit number, $16807=7^5$, is also a fifth power. Similarly, the $9$-digit number, $134217728=8^9$, is a ninth power.

How many $n$-digit positive integers exist which are also an $n$th power?

## べき乗の桁の個数

$5$桁の数 $16807=7^5$は自然数を5乗した数である. 同様に$9$桁の数 $134217728=8^9$も自然数を9乗した数である.

自然数を $n$ 乗して得られる $n$ 桁の正整数は何個あるか?

```scheme

(import (scheme base)

        (gauche base)

        (scheme inexact))

(define (digit-num num)

  (assume (exact-integer? num))

  (assume (not (negative? num)))

  (if (zero? num)

    1

    (+ (floor->exact (log num 10))

       1)))

(define (powerful-digit-list)

  (let loop1 ([base 1]

              [rest (iota 9 1)]

              [result '()])

    (if (null? rest)

      (reverse result)

      (let loop2 ([index (car rest)]

                  [rest rest]

                  [result result])

        (cond

          [(< 9 index)

           (loop1 (+ base 1)

                  rest

                  result)]

          [(< (digit-num (expt index base)) base)

           (loop2 (+ index 1)

                  (delete index rest)

                  result)]

          [(= (digit-num (expt index base)) base)

           (loop2 (+ index 1)

                  rest

                  (cons `(,index . ,base) result))]

          [else

            (loop1 (+ base 1) rest result)])))))

(define answer-63

  (length (powerful-digit-list)))

(format #t "63: ~d~%" answer-63)

```