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I. At first find found all primes <= N (sieve of Eratosthenes). Getting the sum will be easy then.
Follow-up:
Cache the sums for any given N to save time. {N:SUM}
Optimization: Don't have to store sums for every N.
When N = 7, N = 8, N = 9, N = 10, the prime sum remains 17.
For N between 11 to 12, the prime sum is 28.
For N between 13 to 16, the sum is 41.
Use a BST structure as the cache. For N = 16, cache:
{2:3, 4:6, 6:11, 10:17, 12:28, 16:41}

For a given N, call cache.ceilingKey(N) to find the bucket for N.

N/log(n) * log(N)


Complexity
Time:
sieve of Eratosthenes takes O(NloglogN) time.
Insert an element into BST takes O(logN), there are N/logN primes in total to be added.
So building the cache takes logN * N / LogN = O(N) time
requesting primeSum(N) takes O(logN)

Space:
sieve of Eratosthenes takes O(N) extra space which will later be release after the cache is created.
Cache: O(N/logN)

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import java.util.TreeMap;
public class PrimeSum {

    TreeMap<Integer, Integer> sums;

    public PrimeSum(int n) { //input the upper limit for all Ns
        sums = new TreeMap<>();
        // init an array to track prime numbers
        boolean[] primes = new boolean[n + 1];
        for (int i = 2; i < n; i++)
            primes[i] = true;
        for (int i = 2; i <= Math.sqrt(n); i++) {
            if (primes[i]) {
                for (int j = i + i; j < n; j += i)
                    primes[j] = false;
            }
        }
        // insert sums into cache
        int sum = 0;
        for(int i = 2; i <= n; i++) {
            if(primes[i]) {
                sums.put(i - 1, sum);
                sum += i;
            }
        }
        if(primes[n]) {
            sums.put(n, sum);
        }
    }

    public int primeSum(int N) {
        Integer ceiling = sums.ceilingKey(N);
        //if(ceiling == null) {
            //Exception("input value overflows");
        //}
        return sums.get(ceiling);
    }

}

With O(Max_N) preprocessing time and O(Max_N) storage with subsequent queries in O(1) time -

class Primes:
    __primeUptoN = [0, 0]
    __primeSum = [0]

    def __init__(self):
        self.__generatePrimesIfNotGeneratedBefore()

    def primeSum(self, N):
        return self.__primeSum[self.__primeUptoN[N]]

    @classmethod
    def __generatePrimesIfNotGeneratedBefore(Primes):
        if len(Primes.__primeSum) > 1:
            return
        MAX_N = 10 ** 6
        non_primes = set()
        for i in range(2, MAX_N + 1):
            if i not in non_primes:
                Primes.__primeSum.append(Primes.__primeSum[-1] + i)
                j = i + i
                while j <= MAX_N:
                    non_primes.add(j)
                    j += i
            Primes.__primeUptoN.append(len(Primes.__primeSum) - 1)

package general;

import java.util.Arrays;
import java.util.TreeMap;

public class SieveOfEratosthenese {

	public static void getPrimesBySeieveOfEraosthenes(int n, boolean primes[]) {

		for(int i=2; i<= Math.sqrt(n); i++){
			if(primes[i]){
				for(int j=i*i;j<=n; j+=i){
					primes[j] = false;
				}
			}
		}
	}

	public static void main(String[] args) {
		// TODO Auto-generated method stub

		int n = 16;
		boolean[] primes = new boolean[n+1];
		Arrays.fill(primes, true);
		getPrimesBySeieveOfEraosthenes(n, primes);
		
		for (int i = 2; i <= n; i++) {
			if (primes[i])
				System.out.print((i) + " ");
		}
		
		TreeMap <Integer, Integer> sums = new TreeMap<>();
		
		int sum =0;
		
		
		for (int i = 2; i <= n; i++) {
			if (primes[i]){
				sums.put(i-1, sum);
				sum +=i;
			}
			
		}
		
		if(primes[n]){
			sums.put(n,sum);
		}
		
		System.out.println( "\n"+  sums.get(sums.ceilingKey(4)));
	}

}

public class SumOfPrimeNumbers {
    static List<Integer> cachePrime = new ArrayList<>();
    static List<Integer> cacheSumPrime = new ArrayList<>();
    static {
        add(2,2);
        add(3,5);
    }

    static private void add(int n, int sum) {
        cachePrime.add(n);
        cacheSumPrime.add(sum);
    }

    static private int nextPrime(int prime) {
        while( true ) {
            ++prime;
            int i = 0;
            for(  ; i < cachePrime.size() ; ++i )
                if( prime % cachePrime.get(i) == 0 )
                    break;
            if( i == cachePrime.size() )
                return prime;
        }
    }

    static public int sumPrime(int n) {
        if( cachePrime.contains(n) ) {
            return cacheSumPrime.get( cachePrime.indexOf(n) );
        }

        int lastPrime = cachePrime.size()-1;
        if( cachePrime.get(lastPrime) > n ) {
            while (cachePrime.get(lastPrime) > n) {
                --lastPrime;
            }
        }
        else {
            int nextPrime = nextPrime(cachePrime.get(lastPrime));
            while (cachePrime.get(lastPrime) < n && nextPrime < n) {
                add(nextPrime, nextPrime + cacheSumPrime.get(lastPrime));
                lastPrime = cachePrime.size() - 1;
                nextPrime = nextPrime(cachePrime.get(lastPrime));
            }
        }
        return cacheSumPrime.get(lastPrime);
    }
}

import java.util.*;
public class Primes {

	Map<Integer, Integer> cache = new HashMap<Integer, Integer>();
	public static void main(String[] args) {
		Primes primes = new Primes();
		primes.allPrimesTo(20);		
		System.out.println(primes.cache.get(13));
		System.out.println(primes.cache.get(5));
	}
	
	int allPrimesTo(int n) {
		int sum = 0;
		for(int i=2; i<=n; i++) {
			if(isPrime(i)) {
				sum+=i;
				cache.put(i, sum);
			}
		}
		return sum;
	}
	
	boolean isPrime(int n) {
		if(n ==2 ) return false;
		for(int i=2; i<n; i++) {
			if(n%i==0) return false;
		}
		return true;
	}

}