<< problem 203 - Squarefree Binomial Coefficients | Dice Game - problem 205 >> |
Problem 204: Generalised Hamming Numbers
(see projecteuler.net/problem=204)
A Hamming number is a positive number which has no prime factor larger than 5.
So the first few Hamming numbers are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15.
There are 1105 Hamming numbers not exceeding 10^8.
We will call a positive number a generalised Hamming number of type n, if it has no prime factor larger than n.
Hence the Hamming numbers are the generalised Hamming numbers of type 5.
How many generalised Hamming numbers of type 100 are there which don't exceed 10^9?
My Algorithm
After all prime numbers between 2 and hamming
are found, the function search
builds all combinations of prime factors.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "5 100000000" | ./204
Output:
Note: the original problem's input 100 1000000000
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)
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>
// user-defined limits
unsigned int limit = 1000000000;
unsigned int hamming = 100;
// short list of involved primes
std::vector<unsigned int> primes;
// return number of hamming numbers where one prime factor is at least primes[indexMinPrime]
// and which is a multiple of x
unsigned int search(unsigned long long x = 1, unsigned int indexMinPrime = 0)
{
unsigned int result = 1;
// multiply current number by all allowed primes
for (auto i = indexMinPrime; i < primes.size(); i++)
{
auto product = primes[i] * x;
// too large ?
if (product > limit)
break;
// multiply by more primes
result += search(product, i);
}
return result;
}
int main()
{
std::cin >> hamming >> limit;
// the usual prime sieve
for (unsigned int i = 2; i <= hamming; i++)
{
bool isPrime = true;
for (auto p : primes)
{
if (p*p > i)
break;
if (i % p == 0)
{
isPrime = false;
break;
}
}
if (isPrime)
primes.push_back(i);
}
// count hamming numbers
std::cout << search() << std::endl;
}
This solution contains 11 empty lines, 9 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.02 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(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
May 27, 2017 submitted solution
May 27, 2017 added comments
Difficulty
Project Euler ranks this problem at 30% (out of 100%).
Links
projecteuler.net/thread=204 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/hughdbrown/Project-Euler/blob/master/euler-204.py (written by Hugh Brown)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p204.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/204.py (written by Stephen Macke)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/204.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/204.cpp (written by Yuping Luo)
C github.com/LaurentMazare/ProjectEuler/blob/master/e204.c (written by Laurent Mazare)
C github.com/zmwangx/Project-Euler/blob/master/204/204.c (written by Zhiming Wang)
Java github.com/HaochenLiu/My-Project-Euler/blob/master/204.java (written by Haochen Liu)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p204.java (written by Nayuki)
Go github.com/frrad/project-euler/blob/master/golang/Problem204.go (written by Frederick Robinson)
Mathematica github.com/steve98654/ProjectEuler/blob/master/204.nb
Clojure github.com/rm-hull/project-euler/blob/master/src/euler204.clj (written by Richard Hull)
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 203 - Squarefree Binomial Coefficients | Dice Game - problem 205 >> |