PROBLEM LINK:
Authors: Parth Dhorajiya and Rituj Aryan
Tester: Rohit Doshi
DIFFICULTY:
MEDIUMHARD
PREREQUISITES:
Number Theory, Dynamic Programming
PROBLEM:
You are given three integers L, R, and S. Find the total number of subsets wherein the summation of the total number of prime factors of distinct numbers (lying between L and R, both inclusive) is equal to S.
The prime factor of a number is that divisor which has a factor as 1 and that number itself only. Can you help Cheems to solve this question?
EXPLANATION:
Idea behind the problem is :
 Make a list v_1 in which i^{th} element stores the number of prime factors of the (L+i)^{th} element.
 This is classic subset sum problem in which we have to count the number of subsets having sum = S.
For 1) – A number N can be represented in prime factorization as
N = a^p * b^q * c^r * d^s * e^t ...
where (a,b,c,d,e, ... are prime factors in increasing order)
For every number lying in range [L,R]:
Suppose current number is N.
count = 0; // Number of prime factors
for i = 2 to sqrt(N) {
if (N%i == 0) {
count++;
N = N/i;
}
while(N%i == 0) {
N = N/i;
}
if (N > 1) {
count++;
}
Store the count in list v1
(This denotes the number of prime factor of given number N)
}
Then sort v_1 in nondecreasing order.
For 2) –

We make dp[(RL+1)+1][S+1] in which dp[i][j] denotes the number of subsets (considering first i elements of v_1) having sum j.

Initialize dp[i][0]=1 (1st column of dp[ ][ ]) for valid 0 \leq i \leq (RL+1), since there is always an empty subset { } having sum 0.

For every particular element we have 2 options, to include it in the contribution of the sum or not.

If v_1[i1] > j, i.e. if the i^{th} element is greater than j (current sum).
Then its impossible to include it so the total number of subset would be same as of considering (first i1 elements) having sum j. 
If v_1[i1] \leq j, i.e we have option to include it.
So we add dp[i1][j] + dp[i1][jv_1[i1]],
first term means we consider first (i1) elements that sums upto j, and
second term means we consider i^{th} element of v_1 and the sum remaining is (jv_1[i1]). So we would consider first (i1) elements that sums to (jv_1[i1]).


dp[n][s] would give our required answer.
CORNER CASE: (L = 1)
Now since our first element in v_1 would be 0
So we can initialize dp[0][0]=1 and dp[i][0] = 2, 1 \leq i \leq (RL+1)
Since we have 2 subsets having sum 0: ({ }, 0).
OR
Initialize 1st column in dp [ ] [ ] =1 as did before and multiply the answer we would get by 2. (because for every subset we can get a new subset by adding 0 to it.)
SOLUTIONS:
Setter's Solution  CPP
#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define pb push_back
#define mp make_pair
#define A first
#define B second
#define lb long double
#define fast ios_base::sync_with_stdio(false);cin.tie(NULL);
#define fo(i,n) for(i=0;i<n;i++)
#define mpp map<ll,ll>m;
#define vec vector<ll>v;
#define mod (ll)(1e9+7)
ll a[1005][1005];
void solve()
{
ll l,r,s,i,j,c=0,n,count=0;
cin>>l>>r>>s;
vector<ll> v,v1;
for(i=l;i<=r;i++)
v.pb(i);
for (auto e:v) {
ll a=e;
c=0;
for(i=2;i*i<=a;i++)
{
if(a%i==0)
{
c++;
a=a/i;
}
while(a%i==0)
{
a=a/i;
}
}
if(a>1)
c++;
v1.pb(c);
}
sort(v1.begin(),v1.end());
n=v1.size();
ll dp[n+1][s+1];
for(i=0;i<=n;i++)
dp[i][0]=1;
for(j=1;j<=s;j++)
dp[0][j]=0;
for(i=1;i<=n;i++){
for(j=1;j<=s;j++){
if(v1[i1]<=j){
dp[i][j]=(dp[i1][j]%mod+dp[i1][jv1[i1]]%mod)%mod;
dp[i][j]%=mod;
}
else{
dp[i][j]=dp[i1][j]%mod;
dp[i][j]%=mod;
}
}
}
if(l>1)
cout<<dp[n][s]%mod<<endl;
else{
cout<<((dp[n][s]%mod*2)%mod)<<endl;
}
}
int main()
{
fast
solve();
}
Tester's Solution  Python
from math import sqrt
mod = 10**9 + 7
MAXN = 1000001
SPF = [i for i in range(MAXN)]
# Calculating SPF (Smallest Prime Factor) for every number till MAXN.
# Time Complexity : O(n log(logn))
def manipulated_seive():
for i in range(4, MAXN, 2):
SPF[i] = 2
i = 3
while(i*i < MAXN):
if SPF[i] == i:
for j in range(i*i, MAXN, i):
if SPF[j] == j:
SPF[j] = i
i += 1
# A O(log n) function returning primefactorization by dividing by smallest prime factor at every step
def getFactorization(x):
ret = []
while (x != 1):
ret.append(SPF[x])
x = x // SPF[x]
return len(set(ret))
def subsetSum(numbers, sum):
array = [1] + [0] * (sum)
for current_number in numbers:
for num in range(sum  current_number, 1, 1):
if array[num]:
array[num + current_number] = (array[num + current_number]%mod + array[num]%mod)%mod
return array[sum]%mod
manipulated_seive()
l, r, s = map(int, input().split())
primeDiv = []
for i in range(l, r+1):
primeDiv.append(getFactorization(i))
ans = subsetSum(primeDiv, s)%mod
print(ans)
Feel free to share your approach. In case of any doubt or anything is unclear please ask it in the comment section. Any suggestions are welcomed.