<< problem 186 - Connectedness of a network | The hyperexponentiation of a number - problem 188 >> |

# Problem 187: Semiprimes

(see projecteuler.net/problem=187)

A composite is a number containing at least two prime factors. For example, 15 = 3 * 5; 9 = 3 * 3; 12 = 2 * 2 * 3.

There are ten composites below thirty containing precisely two, not necessarily distinct, prime factors: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26.

How many composite integers, n < 10^8, have precisely two, not necessarily distinct, prime factors?

# My Algorithm

My fast prime sieve generates all prime numbers below 5 * 10^7 in about 120ms.

Then all primes numbers are pairwise multiplied. I count the number of all products below 10^8.

## Alternative Approaches

A prime counting function can solve this almost instantly (see my solution for problem 501).

Or even simpler: `prime[j]`

is the smallest prime to be multiplied with `prime[i]`

.

Then `limit / prime[i]`

will be the maximum number for `prime[j]`

. A simple binary search in `primes`

can count how many primes are relevant.

(that's prime counting, too, and equivalent to the speed-up trick in problem 501 for small `n`

).

# My code

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

The code contains `#ifdef`

s to switch between the original problem and the Hackerrank version.

Enable `#ifdef ORIGINAL`

to produce the result for the original problem (default setting for most problems).

#include <iostream>
#include <vector>
#include <algorithm>
// ---------- standard prime sieve from my toolbox ----------

// odd prime numbers are marked as "true" in a bitvector

std::vector<bool> sieve;
// return true, if x is a prime number

bool isPrime(unsigned int x)
{
// handle even numbers
if ((x & 1) == 0)
return x == 2;
// lookup for odd numbers
return sieve[x >> 1];
}
// find all prime numbers from 2 to size

void fillSieve(unsigned int size)
{
// store only odd numbers
const unsigned int half = size >> 1;
// allocate memory
sieve.resize(half, true);
// 1 is not a prime number
sieve[0] = false;
// process all relevant prime factors
for (unsigned int i = 1; 2*i*i < half; i++)
// do we have a prime factor ?
if (sieve[i])
{
// mark all its multiples as false
unsigned int current = 3*i+1;
while (current < half)
{
sieve[current] = false;
current += 2*i+1;
}
}
}
// ---------- and now problem-specific code ----------

int main()
{
// generate all primes
unsigned int limit = 100000000;
unsigned int largestPrime = limit / 2 + 100; // at least one more prime than strictly needed
fillSieve(largestPrime);
// extract all prime numbers from sieve
std::vector<unsigned int> primes = { 2 };
primes.reserve(3002000); // avoid frequent re-allocations
for (unsigned int i = 3; i < largestPrime; i += 2)
if (isPrime(i))
primes.push_back(i);
#define ORIGINAL
#ifndef ORIGINAL
// Hackerrank has several test cases
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
#endif
{
std::cin >> limit;
// compute all products of primes i and j where i <= j
unsigned int count = 0;
for (unsigned int i = 0; primes[i] * primes[i] < limit; i++)
for (unsigned int j = i; primes[i] * primes[j] < limit; j++)
// found one more solutions ...
count++;
std::cout << count << std::endl;
}
return 0;
}

This solution contains 14 empty lines, 18 comments and 6 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 30 | ./187`

Output:

*Note:* the original problem's input `100000000`

__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 0.18 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

Peak memory usage was about 17 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

May 23, 2017 submitted solution

May 23, 2017 added comments

August 18, 2017 modified to solve Hackerrank, too

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler187

My code solves **26** out of **26** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **20%** (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 rarely an option.

# Links

projecteuler.net/thread=187 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

# 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 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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

# Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.

All of my solutions can be used for any purpose and I am in no way liable for any damages caused.

You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.

Thanks for all their endless effort.

<< problem 186 - Connectedness of a network | The hyperexponentiation of a number - problem 188 >> |