c++ - 关于 Euler 的 Totient 函数的 ACM 问题(作业)

Question

我的老师给了我们一个关于数学问题的 acm 问题。我试过了，但得到了 TLE。

问题来了

欧拉的 Totient 函数 φ (n) [有时称为 phi 函数] 用于确定与 n 互质且小于 n 的数的数量。例如1、2、4、5、7、8都小于9且与9互质，φ(9)=6。HG 是 X Y 的老师。有一天，HG 想通过数学游戏教 XY 一些关于欧拉 Totient 函数的知识。也就是说，HG 给出了一个正整数 N，XY 告诉他的主人 2<=n<=N 的值，其中 φ(n) 是最大值。很快HG发现这对于作为狼疮入门者的XY来说似乎有点容易，因为XY通过一个小程序给出了非常快的正确答案。所以HG做了一些改变。这次 XY 会告诉他 2<=n<=N 的值，其中 n/φ(n) 是最大值。这次 XY 遇到了一些困难，因为他没有足够的知识来解决这个问题。现在他需要你的帮助。

输入有T个测试用例(1<=T<=50000)。对于每个测试用例，标准输入包含一行 2 ≤ n ≤ 10^100。

输出对于每个测试用例，应该有单行输出来回答上面提出的问题。

示例输入2个10100

示例输出6个30

这是我的代码。谁能帮助改进它？

#include<iostream>
#include<string>
#include<iomanip>
#include<algorithm>
using namespace std;
class BigNum;
istream& operator>>(istream&,  BigNum&);
ostream& operator<<(ostream&,  BigNum&);
#define MAXN 9999
#define MAXSIZE 1000
#define DLEN 4
class BigNum
{
public:
int a[MAXSIZE];
int len;
public:
BigNum(){len = 1;memset(a,0,sizeof(a));}
BigNum(const int);
BigNum(const char*);
BigNum(const BigNum &); 
BigNum &operator=(const BigNum &);
friend istream& operator>>(istream&,  BigNum&);
friend ostream& operator<<(ostream&,  BigNum&);
BigNum operator+(const BigNum &) const;
BigNum operator-(const BigNum &) const;
BigNum operator*(const BigNum &) const;
BigNum operator/(const int  &) const;
BigNum operator^(const int  &) const;
int    operator%(const int  &) const; 
bool   operator>(const BigNum & T)const;
bool   operator==(const BigNum & T)const;
bool   operator==(const int & t)const;
bool   operator>(const int & t)const;
};
istream& operator>>(istream & in,  BigNum & b)
{
char ch[MAXSIZE*4];
int i = -1;
in>>ch;
int l=strlen(ch);
int count=0,sum=0;
for(i=l-1;i>=0;)
{
sum = 0;
int t=1;
for(int j=0;j<4&&i>=0;j++,i--,t*=10)
{
sum+=(ch[i]-'0')*t;
}
b.a[count]=sum;
count++;
}
b.len =count++;
return in;

}
ostream& operator<<(ostream& out,  BigNum& b)
{
int i; 
cout << b.a[b.len - 1];
for(i = b.len - 2 ; i >= 0 ; i--)
{
cout.width(DLEN);
cout.fill('0');
cout << b.a[i];
}
return out;
}
BigNum::BigNum(const int b)
{
int c,d = b; 
len = 0;
memset(a,0,sizeof(a));
while(d > MAXN)
{
c = d - (d / (MAXN + 1)) * (MAXN + 1);
d = d / (MAXN + 1);  a[len++] = c;
}
a[len++] = d;
}
BigNum::BigNum(const char*s)
{
int t,k,index,l;
memset(a,0,sizeof(a));
l=strlen(s); 
len=l/DLEN;
if(l%DLEN)len++;
index=0;
for(int i=l-1;i>=0;i-=DLEN)
{
t=0;k=i-DLEN+1;
if(k<0)k=0;
for(int j=k;j<=i;j++)
t=t*10+s[j]-'0';
a[index++]=t;
}
}
BigNum::BigNum(const BigNum & T) : len(T.len)
{
int i;
memset(a,0,sizeof(a));
for(i = 0 ; i < len ; i++)  a[i] = T.a[i];
}
BigNum & BigNum::operator=(const BigNum & n)
{
len = n.len;
memset(a,0,sizeof(a));
for(int i = 0 ; i < len ; i++)
a[i] = n.a[i];
return *this;
}
BigNum BigNum::operator+(const BigNum & T) const
{
BigNum t(*this);
int i,big; 
big = T.len > len ? T.len : len;
for(i = 0 ; i < big ; i++)
{
t.a[i] +=T.a[i];
if(t.a[i] > MAXN)
{
t.a[i + 1]++;
t.a[i] -=MAXN+1;
}
}
if(t.a[big] != 0) t.len = big + 1;
else t.len = big; 
return t;
}
BigNum BigNum::operator-(const BigNum & T) const
{ 
int i,j,big;
bool flag;
BigNum t1,t2;
if(*this>T)
{
t1=*this;
t2=T;
flag=0;
}
else
{
t1=T;
t2=*this;
flag=1;
}
big=t1.len;
for(i = 0 ; i < big ; i++)
{
if(t1.a[i] < t2.a[i])
{
j = i + 1;
while(t1.a[j] == 0) j++;
t1.a[j--]--;
while(j > i) t1.a[j--] += MAXN;
t1.a[i] += MAXN + 1 - t2.a[i];
}
else t1.a[i] -= t2.a[i];
}
t1.len = big;
while(t1.a[len - 1] == 0 && t1.len > 1)
{
t1.len--;
big--;
}
if(flag)t1.a[big-1]=0-t1.a[big-1];
return t1;
}
BigNum BigNum::operator*(const BigNum & T) const
{
BigNum ret;
int i,j,up;
int temp,temp1; 
for(i = 0 ; i < len ; i++)
{
up = 0;
for(j = 0 ; j < T.len ; j++)
{
temp = a[i] * T.a[j] + ret.a[i + j] + up;
if(temp > MAXN)
{
temp1 = temp - temp / (MAXN + 1) * (MAXN + 1);
up = temp / (MAXN + 1);
ret.a[i + j] = temp1;
}
else
{
up = 0;
ret.a[i + j] = temp;
}
}
if(up != 0)
ret.a[i + j] = up;
}
ret.len = i + j;
while(ret.a[ret.len - 1] == 0 && ret.len > 1) ret.len--;
return ret;
}
BigNum BigNum::operator/(const int & b) const
{
BigNum ret;
int i,down = 0; 
for(i = len - 1 ; i >= 0 ; i--)
{
ret.a[i] = (a[i] + down * (MAXN + 1)) / b;
down = a[i] + down * (MAXN + 1) - ret.a[i] * b;
}
ret.len = len;
while(ret.a[ret.len - 1] == 0 && ret.len > 1) ret.len--;
return ret;
}
int BigNum::operator %(const int & b) const
{
int i,d=0;
for (i = len-1; i>=0; i--)
{
d = ((d * (MAXN+1))% b + a[i])% b; 
}
return d;
}
BigNum BigNum::operator^(const int & n) const
{
BigNum t,ret(1);
int i;
if(n<0)exit(-1);
if(n==0)return 1;
if(n==1)return *this;
int m=n;
while(m>1)
{
t=*this;
for( i=1;i<<1<=m;i<<=1){
t=t*t;
}
m-=i;
ret=ret*t;
if(m==1)ret=ret*(*this);
}
return ret;
}
bool BigNum::operator>(const BigNum & T) const
{
int ln;
if(len > T.len) return true;
else if(len == T.len)
{
ln = len - 1;
while(a[ln] == T.a[ln] && ln >= 0) ln--;
if(ln >= 0 && a[ln] > T.a[ln]) return true;
else return false;
}
else return false;
}

bool BigNum::operator==(const BigNum & T) const
{
int ln;
if(len != T.len) return false;
else
{
ln = len - 1;
while(a[ln] == T.a[ln] && ln-- );
if(ln < 0) return true;
else return false;
}
}

bool BigNum::operator >(const int & t) const
{
BigNum b(t);
return *this>b;
}

bool BigNum::operator==(const int & t) const
{
BigNum b(t);
return *this==b;
} 

const static int PrimeTable[1230]=
{ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
    37, 41, 43, 47, 53, 59, 61, 67, 71, 73,
    79, 83, 89, 97, 101, 103, 107, 109, 113, 127,
    131, 137, 139, 149, 151, 157, 163, 167, 173, 179,
    181, 191, 193, 197, 199, 211, 223, 227, 229, 233,
    239, 241, 251, 257, 263, 269, 271, 277, 281, 283,
    293, 307, 311, 313, 317, 331, 337, 347, 349, 353,
    359, 367, 373, 379, 383, 389, 397, 401, 409, 419,
    421, 431, 433, 439, 443, 449, 457, 461, 463, 467,
    479, 487, 491, 499, 503, 509, 521, 523, 541, 547,
    557, 563, 569, 571, 577, 587, 593, 599, 601, 607,
    613, 617, 619, 631, 641, 643, 647, 653, 659, 661,
    673, 677, 683, 691, 701, 709, 719, 727, 733, 739,
    743, 751, 757, 761, 769, 773, 787, 797, 809, 811,
    821, 823, 827, 829, 839, 853, 857, 859, 863, 877,
    881, 883, 887, 907, 911, 919, 929, 937, 941, 947,
    953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019,
    1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087,
    1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153,
    1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229,
    1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297,
    1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381,
    1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453,
    1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523,
    1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597,
    1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663,
    1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741,
    1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823,
    1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901,
    1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993,
    1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063,
    2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131,
    2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221,
    2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293,
    2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371,
    2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437,
    2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539,
    2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621,
    2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689,
    2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749,
    2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833,
    2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909,
    2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001,
    3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083,
    3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187,
    3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259,
    3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343,
    3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433,
    3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517,
    3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581,
    3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659,
    3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733,
    3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823,
    3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911,
    3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001,
    4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073,
    4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153,
    4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241,
    4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327,
    4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421,
    4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507,
    4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591,
    4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663,
    4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759,
    4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861,
    4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943,
    4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009,
    5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099,
    5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189,
    5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281,
    5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393,
    5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449,
    5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527,
    5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641,
    5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701,
    5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801,
    5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861,
    5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953,
    5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067,
    6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143,
    6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229,
    6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311,
    6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373,
    6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481,
    6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577,
    6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679,
    6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763,
    6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841,
    6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947,
    6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001,
    7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109,
    7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211,
    7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307,
    7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417,
    7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507,
    7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573,
    7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649,
    7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727,
    7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841,
    7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927,
    7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039,
    8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117,
    8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221,
    8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293,
    8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389,
    8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513,
    8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599,
    8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681,
    8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747,
    8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837,
    8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933,
    8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013,
    9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127,
    9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203,
    9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293,
    9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391,
    9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461,
    9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539,
    9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643,
    9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739,
    9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817,
    9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901,
    9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973, 10007,10009
};

int main(){
    BigNum num;
    int NUM;
    cin >> NUM;
    for (int i = 0; i < NUM; i++){
        cin >> num;
        BigNum temp1(1);
        BigNum temp2(2);
        int n = 0;
        while(!(temp2 > num)){
            temp1 = temp2;
            //cout << temp1 << endl;
            temp2 = temp2 * PrimeTable[n];
            n++;
        }
        cout << temp1 << endl;;
    }
}

Answer 1

你现在的回答已经很聪明了。

您没有解释您的解决方案为何有效，所以我将简要说明一下:

Phi 是乘法的(对于相对质数 a 和 b，phi(a*b) = phi(a)*phi(b) )对于质数 p，phi( p^n) = p^(n-1)*(p-1).

因此 f(n) = n/phi(n) 显然也是乘法的，f(p^n) = p/(p-1)。因此，为了最大化 f(n)，查看包含大于 1 的素数的 n 的值是没有意义的:您不改变 f(n) 的值，只是让 n 变大。

此外，f(p) 会随着 p 的增加而减小，因此给定两个具有相同数量质因数的 n 值，您宁愿拥有较小的质因数也不愿拥有较大的质因数。由此可知，使 f(n) 最大化的 n <= N 是小于或等于 N 的最大“主要阶乘”。

/解释

至于你的问题:你可以做的一个技巧来加快这个速度是预先计算原数(主要阶乘)(只有大约 60 个小于 10^100)而不是在运行时计算它们。