abc400_C 2^a b^2

C.2^a b²

原题链接

Problem Statement

A positive integer X is called a good integer if and only if it satisfies the following condition:

• There exists a pair of positive integers (a,b) such that X = >2^a b².

For example, 400 is a good integer because 400 = 2² 10².

Given a positive integer N, find the number of good integers between 1 and N, inclusive.

Constraints:

• 1 ≤ N ≤ 10¹⁸
• N is an integer

Sample Input:

400

Sample Output:

24

解题思路：

不难想到，
一种思路是固定2^k枚举每个2^k·1²到2^k·x²(x是使2^k·x²小于n的最大数)，然后从k到k+1，累计每个k下的x求和
另一种思路是固定x²,枚举每个2^k·x²到2^k+1·x²,然后从x到x+1,统计每个x下的k求和
不难发现第二种思路在时间复杂度O(\(\sqrt{n}\))不允许且重复子区间难以计算，而第一种O(lgn)只需要保证每次仅计算区间中是奇数的x的个数即可
PS：不难发现非重复子区间的个数可以由2^k其中k的奇偶性划分,保证奇数与偶数不会重复枚举，那只要直接计算(\(\sqrt{n/4}\)+\(\sqrt{n/2}\))之和即可

AC code1:

void solve(){
    ll n;cin>>n;
    ll ans=0;
    for(int i=1;;i++){
        ll t=(1LL<<i);
        if(t>n) break;
        ll s=sqrtl(n/t);
        ans+=(s+1LL)/2LL;
    }
    cout<<ans<<endl;
}

AC code2:

void solve(){
    ll n;cin>>n;
    cout<<(int)sqrtl(n/4)+(int)sqrtl(n/2)<<endl;
}