<< 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?
My 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, theni
is a prime number !) - it's greater than 1 for composite numbers
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
.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "644 3" | ./47
Output:
(this interactive test is still under development, computations will be aborted after one second)
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.
Benchmark
The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an 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=gnu++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 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.
Links
projecteuler.net/thread=47 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
C# www.mathblog.dk/project-euler-47-distinct-prime-factors/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/40-49/problem47.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p047.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/47 Distinct primes factors.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem047.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p047.mathematica (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler047.scala (written by Michael Bayne)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own. Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
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 46 - Goldbach's other conjecture | Self powers - problem 48 >> |