数论常用内容——反素数

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提到反素数，大家可能比较陌生，这里博主我对反素数的了解也不是很深刻，希望能借此机会来总结一下并和大家交流

概念

对于任何正整数n,其约数个数为f(n),如果某个正整数n满足：对任意正整数i(0

性质

性质1、一个反素数的所有质因子必然是从2开始的连续若干个质数，因为反素数是保证约数个数为x的这个数n尽量小

性质2、如果n=2^t1*3^t2*5^t3*…，那么必有t1>t2>t3…

应用

1.给出一个数n，求一个最小的正整数x，使得x的约数个数为n

2.求出1~n中约数最多的这个数（构建搜索树，每层为素因子的i次幂，其中0<=i<=63,一般取63，因为2^64为long long上限，每层碰到所剩数字/该层代表素因子小于上层质因子时不再拓展本枝）

反素数表

int a[42] = {1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,7560,10080,15120,20160,25200,27720,45360,50400,55440,83160,110880,166320,221760,277200,332640,498960,554400,665280,720720,1081080,1441440,2162160};int b[42] = {1,2,3,4,6,8,9,10,12,16,18,20,24,30,32,36,40,48,60,64,72,80,84,90,96,100,108,120,128,144,160,168,180,192,200,216,224,240,256,288,320};

应用示例

求一个最小正整数，使得它的因子个数为n

typedef unsigned long long ULL;const ULL INF = ~0ULL;int p[16] = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53};int n;ULL ans;void dfs(int dept,ULL tmp,int num){    if(num > n) return;    if(num == n && ans > tmp) ans = tmp;    for(int i=1;i<=63;i++){        if(ans / p[dept] < tmp) break;        dfs(dept+1,tmp *= p[dept],num*(i+1));    }}int main(){    while(cin>>n){        ans = INF;        dfs(0,1,1);        cout<<ans<<endl;    }    return 0;}

求n以内的因子数最多的数

typedef unsigned long long ULL;const ULL INF = ~0ULL;int p[16] = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53};ULL ans,n;int best;void dfs(int dept,ULL tmp,int num){    //到叶子结点，返回    if(dept >= 16) return;    //num记录的因子个数，如果遇到更小的，就更新    if(num > best){        best = num;        ans = tmp;    }    //当因子个数相同时，取值最小的    if(num == best && ans > tmp) ans = tmp;    for(int i=1;i<=63;i++){        if(n / p[dept] < tmp) break;        dfs(dept+1,tmp *= p[dept],num*(i+1));    }}int main(){    while(cin>>n){        ans = INF;        best = 0;        dfs(0,1,1);        cout<<ans<<endl;    }    return 0;}

超大数据(10^80)找n以内的约数最多的数并输出因子个数(java实现)

package main;import java.io.*; import java.math.*; import java.util.*; class Node {     private static final int MAXP = 60;     public BigInteger K;     public long F;     public int N;     public int[] A;     public Node() {         K = BigInteger.ZERO;         A = new int[MAXP];     } } public class Main{     private static final int MAXIP = 250;     private static final int MAXP = 60;     private static BigInteger[] prime;     private static void init() {         boolean[] isPrime = new boolean[MAXIP];         for(int i=0;i<MAXIP;++i){             isPrime[i] = true;         }         isPrime[0] = isPrime[1] = false;         for(int i=4;i<MAXIP;i+=2) {             isPrime[i] = false;         }         for(int i=3;i<MAXIP;i+=2) {             for(int j=3;i*j<MAXIP;j+=2) {                 isPrime[i*j] = false;             }         }         prime = new BigInteger[MAXP];         for(int i=0, j=0;i<MAXIP;++i) {             if(isPrime[i]) {                 prime[j++] = BigInteger.valueOf(i);             }         }     }     public static void main(String args[]) {         init();         List<BigInteger> P = new ArrayList<BigInteger>();         BigInteger MP = BigInteger.ZERO;         List<Node> ans = new ArrayList<Node>();         Scanner cin = new Scanner(new BufferedInputStream(System.in));         //Integer cas = new Integer(3);        while(cin.hasNext()) //cas>0{             BigInteger temp = cin.nextBigInteger();             P.add(temp);             if(temp.compareTo(MP) == 1) {                 MP = temp;             }             ans.add(new Node());             //cas--;        }         Map<Long, BigInteger> map = new HashMap<Long, BigInteger>();         Queue<Node> queue = new LinkedList<Node>();         Node origin = new Node();         origin.K = BigInteger.ONE;         origin.F = 1;         origin.N = 0;         queue.add(origin);         map.put(origin.F, origin.K);         while(!queue.isEmpty()) {             Node u = queue.peek();             queue.remove();             BigInteger compare = map.get(u.F);             if(compare != null) {                 if(compare.compareTo(u.K) == -1) {                     continue;                 }             }             for(int i=0;i<P.size();++i) {                 if(u.K.compareTo(P.get(i)) <= 0) {                     if(u.F > ans.get(i).F) {                         ans.get(i).F = u.F;                         ans.get(i).K = u.K;                     }                     else if(u.F == ans.get(i).F) {                         if(u.K.compareTo(ans.get(i).K) == -1) {                             ans.get(i).K = u.K;                         }                     }                 }             }             for(int i=0;i<u.N;++i) {                 Node v = new Node();                 v.K = u.K.multiply(prime[i]);                 if(v.K.compareTo(MP) <= 0) {                     v.F = u.F / (u.A[i] + 1) * (u.A[i] + 2);                     v.N = u.N;                     for(int j=0;j<u.N;++j) {                         v.A[j] = u.A[j];                     }                     ++ v.A[i];                     boolean flag = true;                     compare = map.get(v.F);                     if(compare != null) {                         if(compare.compareTo(v.K) <= 0) {                             flag = false;                         }                         else {                             map.remove(v.F);                         }                     }                     if(flag) {                         queue.add(v);                         map.put(v.F, v.K);                     }                 }             }             Node v = new Node();             v.K = u.K.multiply(prime[u.N]);             if(v.K.compareTo(MP) <= 0) {                 v.F = u.F * 2;                 v.N = u.N + 1;                 for(int i=0;i<u.N;++i) {                     v.A[i] = u.A[i];                 }                 ++ v.A[u.N];                 boolean flag = true;                 compare = map.get(v.F);                 if(compare != null) {                     if(compare.compareTo(v.K) <= 0) {                         flag = false;                     }                     else {                         map.remove(v.F);                     }                 }                 if(flag) {                     queue.add(v);                     map.put(v.F, v.K);                 }             }         }         for(int i=0;i<ans.size();++i) {             System.out.println(ans.get(i).K.toString() + " " + ans.get(i).F);        }     } }

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