You can use a Persistent Segment Tree to store the sum of exponents of Primes. And then for every array element upgrade the Tree with the new sum of exponents.
Your answer would be query(version[r])-query(version[l-1]), taking version[0] to be completely empty.
This question took a long while and a lot of WA/TLE to solve.
Here’s my approach without using trees:
Since X and Y can be a maximum of 10^6, we seive till 10^6 to mark the prime numbers.
Prime factorize every number and store it contigiously in an array, while keeping the index of start of each number in another array.
Divide the entire array into root(n) blocks each of (n) elements.
For each block, calculate the number of each primes (1 to 10^6) and store it as sum in a 2D array with row denoting the block number and columns denoting each prime (2,3,5…)
For every input L R X Y, find the first prime after X and the last prime before Y.
Find the number of blocks that completely lie inside L and R.
Use the following to determine the number of primes between X and Y in the range Block after L to block before R by using the 2D array generated earlier.
For the elements before the first block and after the last block, just iterate through the prime factorization and increment ans if prime is between X and Y (at max root(n) operations.
I took each prime number and found out how many of its multiples exist in the array.
in the given example 2 3 4 5 is the array and first prime number is 2 so at each of the indices that contains a multiple of 2 add the exponent(to an empty array initially)
given array:2 3 4 5
initially: 0 0 0 0- state 0
for 2 the array becomes : 1 0 2 0- state 1
then for 3: 1 1 2 0-state 2
for 5: 1 1 2 1-state 3
each of these is a state so for l,r,x,y such as 1 3 2 3 the answer is sum of numbers from 1 to 3 in state 2 minus that in state 0. I was stuck here then I saw Gaurav sen’s video on persistent segment trees and got the answer in the required time.
Mine solution is a simple one. Just use the sqrt trick for summation between ranges. As in this problem, along with ranges, another query pair (X, Y) is added, maintain prefix sum for it over the sqrt-root ranges. 2 things are needed to be taken care of in this method:
Memory size. Naively is it Max(A_i) * No of Blocks, but it can be compressed to P * No.of.Blocks, where P is number of primes below 10^6.
Choosing an efficient block size would be either of (R, L) pair or (X, Y) pair. My solution timed out for (R, L) pair but passed for (X, Y) pair.
Hey, This problem can be solved using just segment tree, The approach is we can store the prime factors as key and their count as the value in the leaf nodes of segment tree and we can merge the same keys and add up their count. I have written a blog which explains my solution, have a look here. Link to the blog.
We can also view this problem as finding the number of points inside a rectangle.
The points are of the form (X,Y) where X is the index of the number arr[X] in the array and for each X,
Y’s are the Prime factors of arr[X].
The queries are of the form L,R,X,Y which translate to the rectangle formed by x=L,x=R,y=X,y=Y.
So we have points in 2d plane and queries are given by rectangles. we have to find the number of points inside this rectangle(including the border) which can be done by segment trees, Square root decomposition etc.
Is this approach similar to(In general) maintaining a cumulative frequency of primes then binary searching the indices for x and y ? and doing this thing with either Segment tree/square root decomposition ?