Problem 72: Counting fractions

(see projecteuler.net/problem=72)

Consider the fraction, dfrac{n}{d}, where n and d are positive integers. If n<d and HCF(n,d)=1, it is called a reduced proper fraction.

If we list the set of reduced proper fractions for d <= 8 in ascending order of size, we get:

dfrac{1}{8}, dfrac{1}{7}, dfrac{1}{6}, dfrac{1}{5}, dfrac{1}{4}, dfrac{2}{7}, dfrac{1}{3}, dfrac{3}{8}, dfrac{2}{5}, dfrac{3}{7}, dfrac{1}{2}, dfrac{4}{7}, dfrac{3}{5}, dfrac{5}{8}, dfrac{2}{3}, dfrac{5}{7}, dfrac{3}{4}, dfrac{4}{5}, dfrac{5}{6}, dfrac{6}{7}, dfrac{7}{8}

It can be seen that there are 21 elements in this set.

How many elements would be contained in the set of reduced proper fractions for d <= 1,000,000?

Algorithm

All reduced fraction have gcd(numerator, denominator) = 1 (gcd stands for "greatest common divisor")
therefore for a given denominator the number of suitable numerators is the same as the Euler totient of the denominator.

In project 70 I implemented phi(x) like this:
result = x * (1 - 1/prime1) * (1 - 1/prime2) * (1 - 1/prime3) * ...
I realized that something like a prime sieve can do the same job "in reverse" much faster:
- initialize phi(x) = x for all numbers
- for all multiples of a prime: phi_{new}(k * prime) = phi_{old}(k * prime) * (1 - 1/prime) = phi_{old}(k * prime) - phi_{old}(k * prime) / prime
- compute all sums phi(2) + phi(3) + ... + phi(i)

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()
{
// 1. initialize phi(x) = x for all numbers
unsigned int limit = 1000000;
std::vector<unsigned int> phi(limit + 1); // vectors are zero-based
for (size_t i = 0; i < phi.size(); i++)
phi[i] = i;
 
// 2. for all multiples of a prime:
// phi_new(k*prime) = phi_old(k*prime) * (1 - 1/prime)
// = phi_old(k*prime) - phi_old(k*prime) / prime
for (unsigned int i = 2; i <= limit; i++)
{
// prime number ? (because not modified yet by other primes)
if (phi[i] == i)
// adjust all multiples
for (unsigned int k = 1; k * i <= limit; k++)
phi[k * i] -= phi[k * i] / i;
}
 
// note: since we are only interested in 0 < fractions < 1
// we have to exclude phi(1) (which would yield 1/1 = 1)
// and start at phi(2)
 
std::vector<unsigned long long> sums(phi.size(), 0);
// 3. compute all sums phi(2) + phi(3) + ... + phi(i)
for (unsigned int i = 2; i <= limit; i++)
sums[i] = sums[i - 1] + phi[i];
 
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
std::cin >> limit;
std::cout << sums[limit] << std::endl;
}
}

This solution contains 5 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:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

This is equivalent to
echo "1 8" | ./72

Output:

(please click 'Go !')

Note: the original problem's input 1000000 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 0.03 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 13 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

March 13, 2017 submitted solution
May 2, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler072

My code solved 21 out of 21 test cases (score: 100%)

Difficulty

Project Euler ranks this problem at 20% (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=72 - 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-72-reduced-proper-fractions/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p072.java (written by Nayuki)
Go: github.com/frrad/project-euler/blob/master/golang/Problem072.go (written by Frederick Robinson)
Scala: github.com/samskivert/euler-scala/blob/master/Euler072.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.

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The 133 solved problems had an average difficulty of 16.9% at Project Euler and I scored 11,174 points (out of 12300) at Hackerrank's Project Euler+.
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