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

# Problem 47: Distinct primes factors

(see projecteuler.net/problem=47)

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:

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

April 19, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler047

My code solved **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 never an option.

# Links

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