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문제

정수론 중간고사를 마치고 집으로 돌아온 상근이는 패닉에 빠졌다. 유일하게 공부를 하지 않은 것이 오일러 피 함수(Euler's totient function, \(\varphi \))였는데, 그 함수에 관한 문제만 나왔기 때문이다. 상근이는 너무 억울했고, 직접 Totient 함수를 만들기로 했다.

정수론에서 양의 정수의 소인수는 그 정수를 나머지 없이 나눌 수 있는 소수이다. 상근이는 $n \ge 2$에서 함수 $F(n)$을 곱이 $n$이 되는 감소하지 않는 소수의 리스트로 정의했다. 예를 들어, $F(8) = \ll 2,2,2 \gg$, $F(60) = \ll 2,2,3,5 \gg$, $F(71) = \ll 71 \gg$ 이다. $O(n)$은 $F(n)$의 길이이다. 예를 들어, $O(8) = 3$, $O(60) = 4$, $O(71) = 1$ 이 된다. 마지막으로, 양의 정수에 대해서 \(p(n)\)을 다음과 같이 정의했다.

\(p(n) = \begin{cases} 0 & \text{if } n = 1 \\ -1 & \text{if } n \text{ is a prime number} \\ O(n) & \text {otherwise} \end{cases}\)

아래 표에는 \(p(n)\)의 첫 20개 값이 나와있다.

$n$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$ $18$ $19$ $20$
$p(n)$ $0$ $-1$ $-1$ $2$ $-1$ $2$ $-1$ $3$ $2$ $2$ $-1$ $3$ $-1$ $2$ $2$ $4$ $-1$ $3$ $-1$ $3$

$a \le b$를 만족하는 두 양의 정수 $a$와 $b$에 대해서, 상근이는 자신의 Totient 함수인 \(\varphi(a,b)\)를 다음과 같이 정의했다.

\(\varphi (a,b)= ( \sum _{ k=a }^{ b }{ p(k) }  )  - (b-a+1)\)

예를 들어, \(\varphi(1,4) = -4\), \(\varphi(16,16) = 3\), \(\varphi(8,12) = 4\) 이다.

구간 $[L, U]$가 주어졌을 때, 가장 큰 값을 갖는 \(\varphi\)를 찾는 프로그램을 작성하시오.

즉, $L \le U$를 만족하는 두 양의 정수 $L$과 $U$가 주어졌을 때, 가장 큰 \(\varphi(a,b)\) ($L \le a \le b \le U$) 를 찾는 프로그램을 작성하시오. 예를 들어, 구간 $[1,20]$에서 가장 큰 \(\varphi\)는 7이다. (\(\varphi(8,16)\))

입력

입력은 여러 개의 테스트 케이스로 이루어져 있다. 각 테스트 케이스는 한 줄로 이루어져 있고, $L$과 $U$가 주어진다. ($1 \le L \le U < 1,000,000$)

입력의 마지막 줄에는 $-1$이 두 개 주어진다.

출력

각 테스트 케이스마다 주어진 구간 $[L, U]$에서 찾을 수 있는 가장 큰 \(\varphi\) 값을 출력한다.

예제 입력 1

1 5
1 20
10 20
900000 901000
-1 -1

예제 출력 1

1. 1
2. 7
3. 5
4. 2551
[{"problem_id":"4924","problem_lang":"0","title":"\uc815\uc218\ub860 \uc2eb\uc5b4","description":"<p>\uc815\uc218\ub860 \uc911\uac04\uace0\uc0ac\ub97c \ub9c8\uce58\uace0 \uc9d1\uc73c\ub85c \ub3cc\uc544\uc628 \uc0c1\uadfc\uc774\ub294 \ud328\ub2c9\uc5d0 \ube60\uc84c\ub2e4. \uc720\uc77c\ud558\uac8c \uacf5\ubd80\ub97c \ud558\uc9c0 \uc54a\uc740 \uac83\uc774 \uc624\uc77c\ub7ec \ud53c \ud568\uc218(Euler&#39;s totient function, \\(\\varphi \\))\uc600\ub294\ub370, \uadf8 \ud568\uc218\uc5d0 \uad00\ud55c \ubb38\uc81c\ub9cc \ub098\uc654\uae30 \ub54c\ubb38\uc774\ub2e4. \uc0c1\uadfc\uc774\ub294 \ub108\ubb34 \uc5b5\uc6b8\ud588\uace0, \uc9c1\uc811 Totient \ud568\uc218\ub97c \ub9cc\ub4e4\uae30\ub85c \ud588\ub2e4.<\/p>\r\n\r\n<p>\uc815\uc218\ub860\uc5d0\uc11c \uc591\uc758 \uc815\uc218\uc758 \uc18c\uc778\uc218\ub294 \uadf8 \uc815\uc218\ub97c \ub098\uba38\uc9c0 \uc5c6\uc774 \ub098\ub20c \uc218 \uc788\ub294 \uc18c\uc218\uc774\ub2e4. \uc0c1\uadfc\uc774\ub294 $n \\ge 2$\uc5d0\uc11c \ud568\uc218 $F(n)$\uc744 \uacf1\uc774 $n$\uc774 \ub418\ub294 \uac10\uc18c\ud558\uc9c0 \uc54a\ub294 \uc18c\uc218\uc758 \ub9ac\uc2a4\ud2b8\ub85c \uc815\uc758\ud588\ub2e4. \uc608\ub97c \ub4e4\uc5b4, $F(8)&nbsp;= \\ll 2,2,2 \\gg$, $F(60) = \\ll 2,2,3,5 \\gg$, $F(71) = \\ll 71 \\gg$ \uc774\ub2e4. $O(n)$\uc740 $F(n)$\uc758 \uae38\uc774\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4, $O(8) = 3$, $O(60) = 4$, $O(71) = 1$ \uc774 \ub41c\ub2e4. \ub9c8\uc9c0\ub9c9\uc73c\ub85c, \uc591\uc758 \uc815\uc218\uc5d0 \ub300\ud574\uc11c \\(p(n)\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud588\ub2e4.<\/p>\r\n\r\n<p>\\(p(n) = \\begin{cases} 0 &amp; \\text{if } n = 1 \\\\ -1 &amp; \\text{if } n \\text{ is a prime number} \\\\ O(n) &amp; \\text {otherwise} \\end{cases}\\)<\/p>\r\n\r\n<p>\uc544\ub798 \ud45c\uc5d0\ub294 \\(p(n)\\)\uc758 \uccab 20\uac1c \uac12\uc774 \ub098\uc640\uc788\ub2e4.<\/p>\r\n\r\n<table class=\"table table-bordered td-center th-center\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th>$n$<\/th>\r\n\t\t\t<th>$1$<\/th>\r\n\t\t\t<th>$2$<\/th>\r\n\t\t\t<th>$3$<\/th>\r\n\t\t\t<th>$4$<\/th>\r\n\t\t\t<th>$5$<\/th>\r\n\t\t\t<th>$6$<\/th>\r\n\t\t\t<th>$7$<\/th>\r\n\t\t\t<th>$8$<\/th>\r\n\t\t\t<th>$9$<\/th>\r\n\t\t\t<th>$10$<\/th>\r\n\t\t\t<th>$11$<\/th>\r\n\t\t\t<th>$12$<\/th>\r\n\t\t\t<th>$13$<\/th>\r\n\t\t\t<th>$14$<\/th>\r\n\t\t\t<th>$15$<\/th>\r\n\t\t\t<th>$16$<\/th>\r\n\t\t\t<th>$17$<\/th>\r\n\t\t\t<th>$18$<\/th>\r\n\t\t\t<th>$19$<\/th>\r\n\t\t\t<th>$20$<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<th>$p(n)$<\/th>\r\n\t\t\t<td>$0$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>$a \\le b$\ub97c \ub9cc\uc871\ud558\ub294 \ub450 \uc591\uc758 \uc815\uc218 $a$\uc640 $b$\uc5d0 \ub300\ud574\uc11c, \uc0c1\uadfc\uc774\ub294 \uc790\uc2e0\uc758 Totient \ud568\uc218\uc778 \\(\\varphi(a,b)\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud588\ub2e4.<\/p>\r\n\r\n<p>\\(\\varphi (a,b)= ( \\sum _{ k=a }^{ b }{ p(k) } &nbsp;) &nbsp;- (b-a+1)\\)<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, \\(\\varphi(1,4) = -4\\), \\(\\varphi(16,16) = 3\\), \\(\\varphi(8,12) = 4\\) \uc774\ub2e4.<\/p>\r\n\r\n<p>\uad6c\uac04 $[L, U]$\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uac00\uc7a5 \ud070 \uac12\uc744 \uac16\ub294 \\(\\varphi\\)\ub97c \ucc3e\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n\r\n<p>\uc989, $L \\le U$\ub97c \ub9cc\uc871\ud558\ub294 \ub450 \uc591\uc758 \uc815\uc218 $L$\uacfc $U$\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uac00\uc7a5 \ud070 \\(\\varphi(a,b)\\) ($L \\le a \\le b \\le U$) \ub97c \ucc3e\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624. \uc608\ub97c \ub4e4\uc5b4, \uad6c\uac04 $[1,20]$\uc5d0\uc11c \uac00\uc7a5 \ud070 \\(\\varphi\\)\ub294 7\uc774\ub2e4. (\\(\\varphi(8,16)\\))<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \ud55c \uc904\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uace0, $L$\uacfc $U$\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. ($1 \\le L \\le U &lt; 1,000,000$)<\/p>\r\n\r\n<p>\uc785\ub825\uc758 \ub9c8\uc9c0\ub9c9 \uc904\uc5d0\ub294 $-1$\uc774 \ub450 \uac1c \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \uc8fc\uc5b4\uc9c4 \uad6c\uac04 $[L, U]$\uc5d0\uc11c \ucc3e\uc744 \uc218 \uc788\ub294 \uac00\uc7a5 \ud070 \\(\\varphi\\) \uac12\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"4924","problem_lang":"1","title":"Johnny Hates Number Theory","description":"<p>Johnny hates Number Theory! Actually, back in 2002, we came to know that Johnny couldn&rsquo;t count and in 2005 we knew that Johnny couldn&rsquo;t yet add. (But we did know in 2003 that Johnny was street smart enough to solve di\ufb03cult graph problems!) Why Johnny decided to study Number Theory is incomprehensible to us.<\/p>\r\n\r\n<p>Anyhow, back to Johnny. Johnny just failed his comprehensive exam and that was all because of Euler&rsquo;s Totient function (\\(\\varphi\\)). Johnny is so angry that he decides to create his own Totient function. Here&rsquo;s how he described it to his advisor:<\/p>\r\n\r\n<p>In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. Johnny de\ufb01nes function F(n), for $n \\ge 2$, to be the non-decreasing list of prime numbers whose product is $n$. For example, $F(8)&nbsp;= \\ll 2,2,2 \\gg$, $F(60) = \\ll 2,2,3,5 \\gg$, and $F(71) = \\ll 71 \\gg$\u0005 ($71$ is a prime.) Let $O(n)$ be the length of the list $F(n)$ (i.e. its ordinal.) For example, $O(8) = 3$, $O(60) = 4$, and $O(71) = 1$. Johnny also de\ufb01nes function \\(p(n)\\) over positive integers as follows:<\/p>\r\n\r\n<p>\\(p(n) = \\begin{cases} 0 &amp; \\text{if } n = 1 \\\\ -1 &amp; \\text{if } n \\text{ is a prime number} \\\\ O(n) &amp; \\text {otherwise} \\end{cases}\\)<\/p>\r\n\r\n<p>The following table illustrates \\(p(n)\\) for the \ufb01rst twenty positive integers:<\/p>\r\n\r\n<table class=\"table table-bordered td-center th-center\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th>$n$<\/th>\r\n\t\t\t<th>$1$<\/th>\r\n\t\t\t<th>$2$<\/th>\r\n\t\t\t<th>$3$<\/th>\r\n\t\t\t<th>$4$<\/th>\r\n\t\t\t<th>$5$<\/th>\r\n\t\t\t<th>$6$<\/th>\r\n\t\t\t<th>$7$<\/th>\r\n\t\t\t<th>$8$<\/th>\r\n\t\t\t<th>$9$<\/th>\r\n\t\t\t<th>$10$<\/th>\r\n\t\t\t<th>$11$<\/th>\r\n\t\t\t<th>$12$<\/th>\r\n\t\t\t<th>$13$<\/th>\r\n\t\t\t<th>$14$<\/th>\r\n\t\t\t<th>$15$<\/th>\r\n\t\t\t<th>$16$<\/th>\r\n\t\t\t<th>$17$<\/th>\r\n\t\t\t<th>$18$<\/th>\r\n\t\t\t<th>$19$<\/th>\r\n\t\t\t<th>$20$<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<th>$p(n)$<\/th>\r\n\t\t\t<td>$0$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$2$<\/td>\r\n\t\t\t<td>$4$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t\t<td>$-1$<\/td>\r\n\t\t\t<td>$3$<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>Given two positive integers $a$ and $b$ where $a \\le b$, Johnny de\ufb01nes his very own Totient function \\(\\varphi(a,b)\\) as follows:<\/p>\r\n\r\n<p>\\(\\varphi (a,b)= ( \\sum _{ k=a }^{ b }{ p(k) } &nbsp;) &nbsp;- (b-a+1)\\)<\/p>\r\n\r\n<p>For example, \\(\\varphi(1,4) = -4\\), \\(\\varphi(16,16) = 3\\), and \\(\\varphi(8,12) = 4\\).<\/p>\r\n\r\n<p>For his dissertation, Johnny needs a program that determines the maximal \\(\\varphi\\) within a given range $[L, U]$. In other words, given two positive integers $L$, $U$ such that $L \\le U$, the program must \ufb01nd the maximum \\(\\varphi(a, b)\\) where $L \\le a \\le b \\le U$. For example, the maximal \\(\\varphi\\) within the range $[1,20]$ is $7$ (which is \\(\\varphi(8,16)\\).)<\/p>\r\n\r\n<p>Write the program Johnny needs!<\/p>\r\n","input":"<p>Your program will be tested on one or more test cases. Each test case is speci\ufb01ed on a single line. Each test case is speci\ufb01ed using two positive integers $L$ and $U$ separated by one or more spaces, and satisfying the following property: $1 \\le L \\le U &lt; 1,000,000$.<\/p>\r\n\r\n<p>The end of the test cases is indicated by a line made of two $-1$&rsquo;s. That last line is is not part of the test cases.<\/p>\r\n","output":"<p>For each test case, output the result on a single line using the following format:<\/p>\r\n\r\n<pre>\r\nk. result<\/pre>\r\n\r\n<p>Where k is the test case number (starting at 1,) and result is the maximal \\(\\varphi\\) that can be found within the range $[L, U]$.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"English"}]

출처

ICPC > Regionals > Africa and Arab > Arab Collegiate Programming Contest > 2006 Arab Collegiate Programming Contest F번

  • 문제를 번역한 사람: baekjoon
  • 문제의 오타를 찾은 사람: corea