Analysis of past CSP-J preliminary test questions | 2021 CSP-J preliminary test reading program (28-33)

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#include <iostream>
using namespace std;

const int n = 100000;
const int N = n + 1;

int m;
int a[N], b[N], c[N], d[N];  //a[N]标记一个数是不是质数，b[N]用来存质数，c[N]用来存最小质因数的个数，d[N]=p^0+p^1+p^2+...+p^num，p指最大质因子
int f[N], g[N];  //f[N]用来存约数个数，g[N]用来存所有约数之和

void init()
{
    f[1] = g[1] = 1;
    for (int i=2; i<=n; i++) {  //枚举i
        if (!a[i]) { 
            b[m++] = i;  //使用数组b存放地i个质数
            c[i] = 1, f[i] = 2;
            d[i] = 1, g[i] = i + 1;
        }
        for (int j=0; j<m&&b[j]*i<=n; j++) { 
            int k = b[j];  //k为当前的质数
            a[i*k] = 1;  //标记质数k的i倍为合数
            if (i%k==0) {  //遇到i的最小质因数就停止枚举
                c[i*k] = c[i] + 1;  
                f[i*k] = f[i] / c[i*k] * (c[i*k]+1);
                d[i*k] = d[i];
                g[i*k] = g[i] * k + d[i];
                break;
            }
            else {
                c[i*k] = 1;
                f[i*k] = 2 * f[i];
                d[i*k] = g[i];
                g[i*k] = g[i] * (k+1);
            }
        }
    }
}

int main()
{
    init();

    int x;
    cin >> x;
    cout << f[x] << ' ' << g[x] << endl;
    return 0;
}

Assume that the input x is a natural number not exceeding 1000, complete the following true-false and multiple-choice questions:

28. If the input is not "1", deleting line 13 will not affect the output result. ( )

[Answer]: Yes

【Analysis】

Line 13 preprocesses 1. The enumeration starts from 2. There is no data output related to 1 in the whole process. If the input starts from 2, there will be no impact.

29. "f[i]/c[i*k]" in line 25 may be indivisible and rounded down. ( )

[Answer]: Wrong

【Analysis】

f[i] represents the number of prime factors of i, c[i*k] represents the minimum number of prime factors of i*k, f[i]=(num1+1)*(num2+1)*...* (numn+1), where numx is a prime factor, c[i*k]=c[i]+1 (see code)=num1+1, the relationship between these two numbers must be multiples

30. After executing init(), the f array is not monotonically increasing, but the g array is monotonically increasing. ( )

[Answer]: Wrong

【Analysis】

The sum of divisors is not necessarily monotonically increasing, such as g[8]=1+2+4+8=15, g[9]=1+3+9=13

31. The time complexity of the init function is ( )

AO(n)

B.O(nlogn)

C.O(n*n^(1/2))

D.O(n^2)

[Answer]: A

【Analysis】

Characteristics of Euler Linear Screen

32. After executing init(), ( ) among f[1], f[2], f[3]... f[100] is equal to 2.

A.23

B.24

C.25

D.26

[Answer]: C

【Analysis】

Only the number of divisors of prime numbers is 2, so we need to find out how many prime numbers there are in 1-100. There are 25. Choose C.

33. When the input is "1000", the output is ( ).

A.“15 1340”

B.“15 2340”

C.“16 2340”

D.“16 1340”

[Answer]: C

【Analysis】

1000=2^3+5^3, the number of divisors=(3+1)*(3+1)=16. There are two divisors of 1000 and 500, which must exceed 1340, so choose C