A Research Problem（欧拉函数，DFS）

UVA10837

由唯一分解定理：

$$n=\prod\limits_{i=1}^{k}(p_i^{a_i})$$

由欧拉函数定义：

$$\phi(n)\\=n\prod\limits_{i=1}^{k}(1-\frac{1}{p_i})\\=\prod\limits_{i=1}^{k}(p_i^{a_i})\prod\limits_{i=1}^{k}(1-\frac{1}{p_i})\\=\prod\limits_{i=1}^{k}(p_i-1)p_i^{a_i-1}$$

因此，对于给定的$\phi(n)$，我们可以将所有符合条件$m\mod (p_i-1)==0$的素数筛选出来，然后再枚举选取多个$p_i$即可。
但由于素数表只打到10000以内的素数，即$\sqrt {phi_n}$。不过由于大于$\sqrt {phi_n}$的素数至多有一个，所以在特判即可。
因为最后一个素数的范围为$[2,phi_n]$，因此只要这个数与$[2,\sqrt{phi_n}]$内的所有数互质，那么它就是素数。
以下函数为判断最后一个选取的素数。

inline bool Judgt(const int& Remainder) {
	//遍历素数表，判断最后剩下的是否为素数
	for (int i = 0; i < PrimeTable.size(); ++i) {
		const int& prime = PrimeTable[i];
		if (Remainder % prime == 0) {
			return false;
		}	
		if (prime * prime > Remainder) {
			break;
		}
	}
	
	//返回是否未选过
	return Used.find(Remainder) == Used.end();
}

完整代码：

#include <iostream>
#include <algorithm>
#include <cmath>
#include <vector>
#include <cstdio>
#include <cstring>
#include <queue>
#include <string>
#include <unordered_set>
using namespace std;

int m;

bool IsNotPrime[10001];

vector<int> PrimeTable;

vector<int> Candidates;

unordered_set<int> Used;


void InitPrimeTable() {
	for (int i = 2; i <= 10000; ++i) {
		if (IsNotPrime[i] == false) {
			PrimeTable.push_back(i);
			for (int j = i * i; j <= 10000; j += i) {
				IsNotPrime[j] = true;
			}
		}
	}
}

void InitCandidates() {
	for (int i = 0; i < PrimeTable.size(); ++i) {
		const int& prime = PrimeTable[i];
		if (m % (prime - 1) == 0) {
			Candidates.push_back(prime);
		}	
		if ((prime - 1) * (prime - 1) > m) {
			break;
		}
	}
}

int Ans;

inline bool Judgt(const int& Remainder) {
	for (int i = 0; i < PrimeTable.size(); ++i) {
		const int& prime = PrimeTable[i];
		if (Remainder % prime == 0) {
			return false;
		}	
		if (prime * prime > Remainder) {
			break;
		}
	}

	return Used.find(Remainder) == Used.end();
}

void DFS(int Index = 0, int Remainder = m, int n = 1) {
	if (Index == Candidates.size()) {
		if (Remainder == 1) {
			Ans = min(Ans, n);
		}
		else if (Judgt(Remainder + 1)) {
			n *= (Remainder + 1);
			Ans = min(Ans, n);
		}
		return;
	}

	DFS(Index + 1, Remainder, n);

	const int& CurPrime = Candidates[Index];
	
	if (Remainder % (CurPrime - 1)) {
		return;
	}

	Used.insert(CurPrime);

	Remainder /= (CurPrime - 1);
	n *= CurPrime;

	DFS(Index + 1, Remainder, n);

	while (Remainder % CurPrime == 0) {
		Remainder /= CurPrime;
		n *= CurPrime;
		DFS(Index + 1, Remainder, n);
	}

	Used.erase(CurPrime);

}

void InitData() {
	Ans = 200000000;
	Candidates.clear();
        Used.clear();
}

int main() {

	InitPrimeTable();

	int Case = 0;

	while (~scanf("%d", &m) && m) {
		InitData();
		InitCandidates();
		DFS();
		printf("Case %d: %d %d\n", ++Case, m, Ans);
	}

	return 0;

}