Find the prime numbers which can written as sum of most consecutive primes

• Difficulty Level : Medium
• Last Updated : 11 Oct, 2021

Given an array of limits. For every limit, find the prime number which can be written as the sum of the most consecutive primes smaller than or equal to limit.
The maximum possible value of a limit is 10^4.

Example:

Input  : arr[] = {10, 30}
Output : 5, 17
Explanation : There are two limit values 10 and 30.
Below limit 10, 5 is sum of two consecutive primes,
2 and 3. 5 is the prime number which is sum of largest
chain of consecutive below limit 10.

Below limit 30, 17 is sum of four consecutive primes.
2 + 3 + 5 + 7 = 17

Below are steps.

1. Find all prime numbers below a maximum limit (10^6) using Sieve of Sundaram and store them in primes[].
2. Construct a prefix sum array prime_sum[] for all prime numbers in primes[]
prime_sum[i+1] = prime_sum[i] + primes[i].
Difference between two values in prime_sum[i] and prime_sum[j] represents sum of consecutive primes from index i to index j.
3. Traverse two loops , outer loop from i (0 to limit) and inner loop from j (0 to i)
4. For every i, inner loop traverse (0 to i), we check if current sum of consecutive primes (consSum = prime_sum[i] – prime_sum[j]) is prime number or not (we search consSum in prime[] using Binary search).
5. If consSum is prime number then we update the result if the current length is more than length of current result.

Below is implementation of above steps.

C++

 \// C++ program to find Longest Sum of consecutive// primes#includeusing namespace std;const int MAX  = 10000; // utility function for sieve of sundaramvoid sieveSundaram(vector &primes){    // In general Sieve of Sundaram, produces primes smaller    // than (2*x + 2) for a number given number x. Since    // we want primes smaller than MAX, we reduce MAX to half    // This array is used to separate numbers of the form    // i+j+2ij from others where 1 <= i <= j    bool marked[MAX/2 + 1] = {0};     // Main logic of Sundaram. Mark all numbers which    // do not generate prime number by doing 2*i+1    for (int i=1; i<=(sqrt(MAX)-1)/2; i++)        for (int j=(i*(i+1))<<1; j<=MAX/2; j=j+2*i+1)            marked[j] = true;     // Since 2 is a prime number    primes.push_back(2);     // Print other primes. Remaining primes are of the    // form 2*i + 1 such that marked[i] is false.    for (int i=1; i<=MAX/2; i++)        if (marked[i] == false)            primes.push_back(2*i + 1);} // function find the prime number which can be written// as the sum of the most consecutive primesint LSCPUtil(int limit, vector &prime, long long int sum_prime[]){    // To store maximum length of consecutive primes that can    // sum to a limit    int max_length = -1;     // The prime number (or result) that can be represented as    // sum of maximum number of primes.    int prime_number = -1;     // Consider all lengths of consecutive primes below limit.    for (int i=0; prime[i]<=limit; i++)    {        for (int j=0; j limit)                break;             // sum_prime[i]-sum_prime[j] is prime number or not            long long int consSum  = sum_prime[i] - sum_prime[j];             // Check if sum of current length of consecutives is            // prime or not.            if (binary_search(prime.begin(), prime.end(), consSum))            {                // update the length and prime number                if (max_length < i-j+1)                {                    max_length = i-j+1;                    prime_number = consSum;                }            }        }    }     return prime_number;} // Returns the prime number that can written as sum// of longest chain of consecutive primes.void LSCP(int arr[], int n){    // Store prime number in vector    vector primes;    sieveSundaram(primes);     long long int sum_prime[primes.size() + 1];     // Calculate sum of prime numbers and store them    // in sum_prime array. sum_prime[i] stores sum of    // prime numbers from primes to primes[i-1]    sum_prime = 0;    for (int i = 1 ; i <= primes.size(); i++)        sum_prime[i] = primes[i-1] + sum_prime[i-1];     // Process all queries one by one    for (int i=0; i

Java

 // Java program to find longest sum// of consecutive primesimport java.util.*; class GFG{      static int MAX = 10000;  // Store prime number in vectorstatic ArrayList

C#

 // C# program to find longest sum// of consecutive primesusing System;using System.Collections; class GFG{     static int MAX = 10000; // Store prime number in vectorstatic ArrayList primes = new ArrayList(); // Utility function for sieve of sundaramstatic void sieveSundaram(){         // In general Sieve of Sundaram,    // produces primes smaller than    // (2*x + 2) for a number given    // number x. Since we want primes    // smaller than MAX, we reduce MAX    // to half. This array is used to    // separate numbers of the form    // i+j+2ij from others where 1 <= i <= j    bool []marked = new bool[MAX / 2 + 1];    Array.Fill(marked, false);      // Main logic of Sundaram. Mark    // all numbers which do not    // generate prime number by    // doing 2*i+1    for(int i = 1;            i <= (Math.Sqrt(MAX) - 1) / 2; i++)        for(int j = (i * (i + 1)) << 1;                j <= MAX / 2;                j = j + 2 * i + 1)            marked[j] = true;      // Since 2 is a prime number    primes.Add(2);      // Print other primes. Remaining    // primes are of the form    // 2*i + 1 such that marked[i] is false.    for(int i = 1; i <= MAX / 2; i++)        if (marked[i] == false)            primes.Add(2 * i + 1);}  // Function find the prime number// which can be written as the// sum of the most consecutive primesstatic int LSCPUtil(int limit, long []sum_prime){         // To store maximum length of    // consecutive primes that can    // sum to a limit    int max_length = -1;      // The prime number (or result)    // that can be represented as    // sum of maximum number of primes.    int prime_number = -1;      // Consider all lengths of    // consecutive primes below limit.    for(int i = 0; (int)primes[i] <= limit; i++)    {        for(int j = 0; j < i; j++)        {                         // If we cross the limit, then            // break the loop            if (sum_prime[i] - sum_prime[j] >                limit)                break;              // sum_prime[i]-sum_prime[j] is            // prime number or not            long consSum  = sum_prime[i] -                            sum_prime[j];                         int[] prime = (int[])primes.ToArray(typeof(int));                         // Check if sum of current length            // of consecutives is prime or not.            if (Array.BinarySearch(prime,                (int)consSum) >= 0)            {                                 // Update the length and prime number                if (max_length < i - j + 1)                {                    max_length = i - j + 1;                    prime_number = (int)consSum;                }            }        }    }    return prime_number;}  // Returns the prime number that// can written as sum of longest// chain of consecutive primes.static void LSCP(int []arr, int n){    sieveSundaram();      long []sum_prime = new long[primes.Count + 1];      // Calculate sum of prime numbers    // and store them in sum_prime    // array. sum_prime[i] stores sum    // of prime numbers from    // primes to primes[i-1]    sum_prime = 0;    for(int i = 1; i <= primes.Count; i++)        sum_prime[i] = (int)primes[i - 1] +                         sum_prime[i - 1];      // Process all queries one by one    for(int i = 0; i < n; i++)      Console.Write(LSCPUtil(          arr[i], sum_prime) + " ");} // Driver codepublic static void Main(string []arg){    int []arr = { 10, 30, 40, 50, 1000 };    int n = arr.Length;         LSCP(arr, n);}} // This code is contributed by rutvik_56

Output:

5 17 17 41 953

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