Project Euler

[ホーム](https://redmine.noppi.jp) - [[Wiki|Project Euler]]

# [[Problem 12]]

## Highly Divisible Triangular Number

The sequence of triangle numbers is generated by adding the natural numbers. So the $7$<sup>th</sup> triangle number would be $1 + 2 + 3 + 4 + 5 + 6 + 7 = 28$. The first ten terms would be:

$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$

Let us list the factors of the first seven triangle numbers:

 **1:** 1

 **3:** 1,3

 **6:** 1,2,3,6

**10:** 1,2,5,10

**15:** 1,3,5,15

**21:** 1,3,7,21

**28:** 1,2,4,7,14,28

We can see that $28$ is the first triangle number to have over five divisors.

What is the value of the first triangle number to have over five hundred divisors?

## 高度整除三角数

三角数の数列は自然数の和で表わされ, 7番目の三角数は 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 である. 三角数の最初の10項は:

$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$$

となる.

最初の7項について, その約数を列挙すると, 以下のとおり.

 **1:** 1

 **3:** 1,3

 **6:** 1,2,3,6

**10:** 1,2,5,10

**15:** 1,3,5,15

**21:** 1,3,7,21

**28:** 1,2,4,7,14,28

これから, 7番目の三角数である28は, 5個より多く約数をもつ最初の三角数であることが分かる.

では, 500個より多く約数をもつ最初の三角数はいくつか.

```scheme

#!r6rs

#!chezscheme

(import (chezscheme))

(define (triangular-numbers end-condition)

  (let loop ([i 1] [current 0] [result '()])

    (if (and

          (not (null? result))

          (end-condition result))

      result

      (let ([next (+ current i)])

        (loop (add1 i) next (cons next result))))))

(define (divisors num)

  (let ([check-max (isqrt num)])

    (let loop ([i 1] [result '()])

      (cond

        [(< check-max i) result]

        [(zero? (mod num i))

         (loop (add1 i) (cons* i (div num i) result))]

        [else (loop (add1 i) result)]))))

(define over500-divisors-triangular-number

  (triangular-numbers

    (lambda (lis)

      (< 500 (length (divisors (car lis)))))))

(define answer-12

  (car over500-divisors-triangular-number))

(printf "12: ~D~%" answer-12)

```