<< problem 46 - Goldbach's other conjecture Self powers - problem 48 >>

# Problem 47: Distinct primes factors

The first two consecutive numbers to have two distinct prime factors are:
14 = 2 * 7
15 = 3 * 5

The first three consecutive numbers to have three distinct prime factors are:
644 = 2^2 * 7 * 23
645 = 3 * 5 * 43
646 = 2 * 17 * 19

Find the first four consecutive integers to have four distinct prime factors each.
What is the first of these numbers?

# Algorithm

Most of my prime sieves are based on trial division - this time the Sieve of Eratosthenes is a perfect fit.
But instead of "crossing off" composite numbers, I increment a counter:
- it's 1 if the number i itself is the only prime factor (well, then i is a prime number !)
- it's greater than 1 for composite numbers

A counter currentRun is incremented by one as long as the current number has the desired number of distinct prime factors.
If the number of prime factors deviated, then currentRun is reset to zero.

If currentRun >= consecutive then I print the position of the first number, which was i - consecutive + 1.

# 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>

int main()
{
unsigned int maxNumber = 500000;
unsigned int consecutive = 4;
std::cin >> maxNumber >> consecutive;

// a group of consecutive numbers might extend beyond maxNumber
// therefore adjust our search range accordingly
maxNumber += consecutive - 1;

// count prime factors of each number (1 is not considered a prime factor)
std::vector<unsigned int> primeFactors(maxNumber + 1, 0);

// modified Sieve of Eratosthenes
for (unsigned int i = 2; i <= maxNumber; i++)
// is i a prime ?
if (primeFactors[i] == 0)
// all multiples of i have i as a prime factor
for (unsigned int j = i; j <= maxNumber; j += i)
primeFactors[j]++;

// iterate over all numbers
unsigned int currentRun = 0;
for (unsigned int i = 2; i <= maxNumber; i++)
{
// match ?
if (primeFactors[i] == consecutive)
{
currentRun++;

// enough such numbers in a row ? print the first
if (currentRun >= consecutive)
std::cout << (i - consecutive + 1) << std::endl;
}
else
{
// reset counter
currentRun = 0;
}
}

return 0;
}


This solution contains 7 empty lines, 10 comments and 2 preprocessor commands.

# Interactive test

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

Input data (separated by spaces or newlines):

This is equivalent to
echo "644 3" | ./47

Output:

(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 less than 0.01 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 4 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 26, 2017 submitted solution

# Hackerrank

My code solves 15 out of 15 test cases (score: 100%)

# Difficulty

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

Hackerrank describes this problem as easy.

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.

projecteuler.net/thread=47 - the best forum on the subject (note: you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-47-distinct-prime-factors/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p047.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p047.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/40-49/problem47.c (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem047.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/47 Distinct primes factors.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler047.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 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.

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

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The 206 solved problems had an average difficulty of 27.5% at Project Euler and
I scored 12,626 points (out of 14300 possible points, top rank was 20 out ouf ≈60000 in July 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.

 << problem 46 - Goldbach's other conjecture Self powers - problem 48 >>
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