# Count Primes in Ranges

• Difficulty Level : Medium
• Last Updated : 09 Jun, 2022

Given a range [L, R], we need to find the count of total numbers of prime numbers in the range [L, R] where 0 <= L <= R < 10000. Consider that there are a large number of queries for different ranges.
Examples:

```Input : Query 1 : L = 1, R = 10
Query 2 : L = 5, R = 10
Output : 4
2
Explanation
Primes in the range L = 1 to R = 10 are
{2, 3, 5, 7}. Therefore for query, answer
is 4 {2, 3, 5, 7}.
For the second query, answer is 2 {5, 7}.```

A simple solution is to do the following for every query [L, R]. Traverse from L to R, check if current number is prime. If yes, increment the count. Finally, return the count.
An efficient solution is to use Sieve of Eratosthenes to find all primes up to the given limit. Then we compute a prefix array to store counts till every value before limit. Once we have a prefix array, we can answer queries in O(1) time. We just need to return prefix[R] – prefix[L-1].

## C++

 `// CPP program to answer queries for count of``// primes in given range.``#include ``using` `namespace` `std;` `const` `int` `MAX = 10000;` `// prefix[i] is going to store count of primes``// till i (including i).``int` `prefix[MAX + 1];` `void` `buildPrefix()``{``    ``// Create a boolean array "prime[0..n]". A``    ``// value in prime[i] will finally be false``    ``// if i is Not a prime, else true.``    ``bool` `prime[MAX + 1];``    ``memset``(prime, ``true``, ``sizeof``(prime));` `    ``for` `(``int` `p = 2; p * p <= MAX; p++) {` `        ``// If prime[p] is not changed, then``        ``// it is a prime``        ``if` `(prime[p] == ``true``) {` `            ``// Update all multiples of p``            ``for` `(``int` `i = p * 2; i <= MAX; i += p)``                ``prime[i] = ``false``;``        ``}``    ``}` `    ``// Build prefix array``    ``prefix[0] = prefix[1] = 0;``    ``for` `(``int` `p = 2; p <= MAX; p++) {``        ``prefix[p] = prefix[p - 1];``        ``if` `(prime[p])``            ``prefix[p]++;``    ``}``}` `// Returns count of primes in range from L to``// R (both inclusive).``int` `query(``int` `L, ``int` `R)``{``    ``return` `prefix[R] - prefix[L - 1];``}` `// Driver code``int` `main()``{``    ``buildPrefix();` `    ``int` `L = 5, R = 10;``    ``cout << query(L, R) << endl;` `    ``L = 1, R = 10;``    ``cout << query(L, R) << endl;` `    ``return` `0;``}`

## Java

 `// Java program to answer queries for``// count of primes in given range.``import` `java.util.*;` `class` `GFG {``    ` `static` `final` `int` `MAX = ``10000``;` `// prefix[i] is going to store count``// of primes till i (including i).``static` `int` `prefix[] = ``new` `int``[MAX + ``1``];` `static` `void` `buildPrefix() {``    ` `    ``// Create a boolean array "prime[0..n]". A``    ``// value in prime[i] will finally be false``    ``// if i is Not a prime, else true.``    ``boolean` `prime[] = ``new` `boolean``[MAX + ``1``];``    ``Arrays.fill(prime, ``true``);` `    ``for` `(``int` `p = ``2``; p * p <= MAX; p++) {` `    ``// If prime[p] is not changed, then``    ``// it is a prime``    ``if` `(prime[p] == ``true``) {` `        ``// Update all multiples of p``        ``for` `(``int` `i = p * ``2``; i <= MAX; i += p)``        ``prime[i] = ``false``;``    ``}``    ``}` `    ``// Build prefix array``    ``prefix[``0``] = prefix[``1``] = ``0``;``    ``for` `(``int` `p = ``2``; p <= MAX; p++) {``    ``prefix[p] = prefix[p - ``1``];``    ``if` `(prime[p])``        ``prefix[p]++;``    ``}``}` `// Returns count of primes in range``// from L to R (both inclusive).``static` `int` `query(``int` `L, ``int` `R)``{``    ``return` `prefix[R] - prefix[L - ``1``];``}` `// Driver code``public` `static` `void` `main(String[] args) {``    ` `    ``buildPrefix();``    ``int` `L = ``5``, R = ``10``;``    ``System.out.println(query(L, R));` `    ``L = ``1``; R = ``10``;``    ``System.out.println(query(L, R));``}``}` `// This code is contributed by Anant Agarwal.`

## Python3

 `# Python3 program to answer queries for``# count of primes in given range.``MAX` `=` `10000` `# prefix[i] is going to``# store count of primes``# till i (including i).``prefix ``=``[``0``]``*``(``MAX` `+` `1``)` `def` `buildPrefix():``    ` `    ``# Create a boolean array value in``    ``# prime[i] will "prime[0..n]". A``    ``# finally be false if i is Not a``    ``# prime, else true.``    ``prime ``=` `[``1``]``*``(``MAX` `+` `1``)` `    ``p ``=` `2``    ``while``(p ``*` `p <``=` `MAX``):` `        ``# If prime[p] is not changed,``        ``# then it is a prime``        ``if` `(prime[p] ``=``=` `1``):` `            ``# Update all multiples of p``            ``i ``=` `p ``*` `2``            ``while``(i <``=` `MAX``):``                ``prime[i] ``=` `0``                ``i ``+``=` `p``        ``p``+``=``1` `    ``# Build prefix array``    ``# prefix[0] = prefix[1] = 0;``    ``for` `p ``in` `range``(``2``,``MAX``+``1``):``        ``prefix[p] ``=` `prefix[p ``-` `1``]``        ``if` `(prime[p]``=``=``1``):``            ``prefix[p]``+``=``1` `# Returns count of primes``# in range from L to``# R (both inclusive).``def` `query(L, R):``    ``return` `prefix[R]``-``prefix[L ``-` `1``]` `# Driver code``if` `__name__``=``=``'__main__'``:``    ``buildPrefix()` `    ``L ``=` `5``    ``R ``=` `10``    ``print``(query(L, R))` `    ``L ``=` `1``    ``R ``=` `10``    ``print``(query(L, R))` `# This code is contributed by mits.`

## C#

 `// C# program to answer``// queries for count of``// primes in given range.``using` `System;` `class` `GFG``{``static` `int` `MAX = 10000;` `// prefix[i] is going``// to store count of``// primes till i (including i).``static` `int``[] prefix = ``new` `int``[MAX + 1];` `static` `void` `buildPrefix()``{``    ` `    ``// Create a boolean array``    ``// "prime[0..n]". A value``    ``// in prime[i] will finally``    ``// be false if i is Not a``    ``// prime, else true.``    ``bool``[] prime = ``new` `bool``[MAX + 1];` `    ``for` `(``int` `p = 2;``             ``p * p <= MAX; p++)``    ``{` `    ``// If prime[p] is``    ``// not changed, then``    ``// it is a prime``    ``if` `(prime[p] == ``false``)``    ``{` `        ``// Update all``        ``// multiples of p``        ``for` `(``int` `i = p * 2;``                 ``i <= MAX; i += p)``        ``prime[i] = ``true``;``    ``}``    ``}` `    ``// Build prefix array``    ``prefix[0] = prefix[1] = 0;``    ``for` `(``int` `p = 2; p <= MAX; p++)``    ``{``        ``prefix[p] = prefix[p - 1];``        ``if` `(prime[p] == ``false``)``            ``prefix[p]++;``    ``}``}` `// Returns count of primes``// in range from L to R``// (both inclusive).``static` `int` `query(``int` `L, ``int` `R)``{``    ``return` `prefix[R] -``           ``prefix[L - 1];``}` `// Driver code``public` `static` `void` `Main()``{``    ``buildPrefix();``    ``int` `L = 5, R = 10;``    ``Console.WriteLine(query(L, R));` `    ``L = 1; R = 10;``    ``Console.WriteLine(query(L, R));``}``}` `// This code is contributed``// by mits.`

## PHP

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## Javascript

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Output:

```2
4```

Time Complexity: O(n*log(log(n)))

Auxiliary Space: O(n)

Here, n is the size of the prime array, Which is MAX here

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