Codeforces Round #376 (Div. 2) F. Video Cards 数论+数据结构+前缀和

F. Video Cards

time limit per test

1 second

memory limit per test

256 megabytes

input

standard input

output

standard output

Little Vlad is fond of popular computer game Bota-2. Recently, the developers announced the new add-on named Bota-3. Of course, Vlad immediately bought only to find out his computer is too old for the new game and needs to be updated.

There are n video cards in the shop, the power of the i-th video card is equal to integer value ai. As Vlad wants to be sure the new game will work he wants to buy not one, but several video cards and unite their powers using the cutting-edge technology. To use this technology one of the cards is chosen as the leading one and other video cards are attached to it as secondary. For this new technology to work it's required that the power of each of the secondary video cards is divisible by the power of the leading video card. In order to achieve that the power of any secondary video card can be reduced to any integer value less or equal than the current power. However, the power of the leading video card should remain unchanged, i.e. it can't be reduced.

Vlad has an infinite amount of money so he can buy any set of video cards. Help him determine which video cards he should buy such that after picking the leading video card and may be reducing some powers of others to make them work together he will get the maximum total value of video power.

Input

The first line of the input contains a single integer n (1 ≤ n ≤ 200 000) — the number of video cards in the shop.

The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 200 000) — powers of video cards.

Output

The only line of the output should contain one integer value — the maximum possible total power of video cards working together.

Examples

input

4
3 2 15 9

output

27

input

4
8 2 2 7

output

18

Note

In the first sample, it would be optimal to buy video cards with powers 3, 15 and 9. The video card with power 3 should be chosen as the leading one and all other video cards will be compatible with it. Thus, the total power would be 3 + 15 + 9 = 27. If he buys all the video cards and pick the one with the power 2 as the leading, the powers of all other video cards should be reduced by 1, thus the total power would be 2 + 2 + 14 + 8 = 26, that is less than 27. Please note, that it's not allowed to reduce the power of the leading video card, i.e. one can't get the total power 3 + 1 + 15 + 9 = 28.

In the second sample, the optimal answer is to buy all video cards and pick the one with the power 2 as the leading. The video card with the power 7 needs it power to be reduced down to 6. The total power would be 8 + 2 + 2 + 6 = 18.

Source

Codeforces Round #376 (Div. 2)

My Solution

//这题虽然是F题，但属于Div.2 D题难道，所以归类于 Div.2 D(或 Div.1 B)了

数论+数据结构+前缀和

就是以前用树状数组的感觉，比如有一个数x，就在以x为下标的地方插入一个 1，如果删除这个就在下标为x的地方插入一个-1， 这样要查询小于等于k的数的个数只要get(k)就好。

这里不需要维护，所以只要用一个前缀和数组，就可以直接预处理出来。

即对于每个a[i]就在 sum[a[i]]++;

读入完以后预处理出前缀和，for i = 1 ~ maxa,  sum[i] += sum[i - 1];

然后扫一遍i = 1 ~ maxa，对于每个i在a[maxn]数组中出现过的i,(即 sum[i] - sum[i - 1] > 0)的地方，

for(j = 1; j <= maxa; j += i)                                                      //每次处理从 j 到 j + i - 1 的数，这些数都会作为 j 加到 s 里去 

   每次 求出 这个区间上的右端点 r = min(maxa, j + i - 1); //因为最后一次的右端点可能不是 j + i - 1;

   然后 s += j * (sum[r] - sum[j - 1]);                                      //从 j 到 j + i - 1 的数的个数

处理完后，刷新 ans = max(ans, s);

复杂度 O(sigma(maxa / i))  //这里大概是 sigma(maxa / i == 2.5e6

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cstring>
using namespace std;
typedef long long LL;
const int maxn = 2e5 + 8;

LL a[maxn], sum[maxn];

int main()
{
    #ifdef LOCAL
    int t = 0;
    for(int i  = 1; i < (int)2e5; i++){
        t += ((int)2e5) / i;
    }
    cout << "O(sigma(maxa / i)) == " << t << "\n" << endl;
    freopen("f.txt", "r", stdin);
    //freopen("d.out", "w", stdout);
    int T = 4;
    while(T--){
    #endif // LOCAL
    ios::sync_with_stdio(false); cin.tie(0);

    LL n, ans = 0;
    cin >> n;
    for(int i = 0; i < n; i++){
        cin >> a[i];
        sum[a[i]]++;
    }


    LL i, j, s, r, maxa = 2e5;
    for(i = 1; i <= maxa; i++) sum[i] += sum[i - 1];
    sort(a, a + n);
    for( i = 1; i <= maxa; i++){
        if(sum[i] - sum[i - 1]){
            s = 0;
            for(j = i; j <= maxa; j += i){
                if(j > a[n - 1]) break;
                r = min(maxa, j + i - 1);
                s += j * (sum[r] - sum[j - 1]);
            }
            ans = max(ans, s);
        }
    }

    cout << ans << endl;

    #ifdef LOCAL
    memset(sum, 0, sizeof sum);
    cout << endl;
    }
    #endif // LOCAL
    return 0;
}

  Thank you!

                                                                                                                                               ------from ProLights