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# Problem 10: Summation of primes

(see projecteuler.net/problem=10)

The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.

Find the sum of all the primes below two million.

# Algorithm

The prime sieve is more or less unchanged from problem 7.

Then I create a lookup table `sums`

which contains for each prime number p

the sum of all prime numbers <=p.

The test cases may contain numbers which are not prime, too.

Therefore I use `upper_bound`

to find the smallest entry which >=p.

And since we that entry "is one step too far", I go back to the previous entry and print it.

## Modifications by HackerRank

My 2-step design was heavily influenced by Hackerrank's large number of test cases:

the "expensive" precomputation is done once and the test cases are computationally very "cheap".

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <vector>
#include <map>
int main()
{
// prime numbers beyond this are not relevant for the problem
const unsigned int MaxPrime = 2000000;
// precompute all relevant prime numbers
std::vector<unsigned int> primes;
// the only even prime
primes.push_back(2);
// now check all odd numbers for primality
for (unsigned int i = 3; i <= MaxPrime; i += 2)
{
bool isPrime = true;
for (auto p : primes)
{
// no larger prime factor possible ?
if (p*p > i)
break;
// no prime number ?
if (i % p == 0)
{
isPrime = false;
break;
}
}
// yes, we have a new prime
if (isPrime)
primes.push_back(i);
}
// prime numbers were found in ascending order,
// let's add their value and store in a map such that
// [prime number] => [sum of all prime numbers up to the current]
// note: long long is required to avoid overflows
std::map<unsigned int, unsigned long long> sums;
unsigned long long sum = 0;
for (auto p : primes)
{
sum += p;
sums[p] = sum;
}
// the test cases are more or less "smart" lookups
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int x;
std::cin >> x;
// find the closest prime number which is bigger than the input
auto i = sums.upper_bound(x);
// go back to the closest prime number which is smaller than the input
i--;
// show the sum associated to that prime number
std::cout << i->second << std::endl;
}
return 0;
}

This solution contains 8 empty lines, 15 comments and 3 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This is equivalent to`echo "1 10" | ./10`

Output:

*Note:* the original problem's input `2000000`

__cannot__ be entered

because just copying results is a soft skill reserved for idiots.

*(this interactive test is still under development, computations will be aborted after one second)*

# Benchmark

The correct solution to the original Project Euler problem was found in **0.22** seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

Peak memory usage was about 12 MByte.

(compiled for x86_64 / Linux, GCC flags: `-O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL`

)

See here for a comparison of all solutions.

**Note:** interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without `-DORIGINAL`

.

# Changelog

February 23, 2017 submitted solution

March 29, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler010

My code solved **8** out of **8** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **5%** (out of 100%).

Hackerrank describes this problem as **medium**.

*Note:*

Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.

In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

# Similar problems at Project Euler

Problem 7: 10001st prime

*Note:* I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.

# Links

projecteuler.net/thread=10 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/sum-of-all-primes-below-2000000-problem-10/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p010.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p010.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p010.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/10-19/problem10.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem010.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/10 Summation of primes.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler010.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.

yellow problems score less than 100% at Hackerrank (but still solve the original problem).

gray problems are already solved but I haven't published my solution yet.

blue problems are already solved and there wasn't a Hackerrank version of it (at the time I solved it) or I didn't care about it because it differed too much.

*Please click on a problem's number to open my solution to that problem:*

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