Problem 69: Totient maximum

(see projecteuler.net/problem=69)

Euler's Totient function, phi(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n.
For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, phi(9)=6.

nRelatively Primephi(n)n/phi(n)
    
2112
31,221.5
41,322
51,2,3,441.25
61,523
71,2,3,4,5,661.1666...
81,3,5,742
91,2,4,5,7,861.5
101,3,7,942.5

It can be seen that n=6 produces a maximum n/phi(n) for n <= 10.

Find the value of n <= 1000000 for which n/phi(n) is a maximum.

Algorithm

I have no formal proof yet (it's too late - time to go to bed !):
in my first brute-force attempt I observed that the "best number" is the product of all primes
best = 2 * 3 * 5 * 7 * 11 * 13 * ... where best < 1000000.

Simple tests showed that all primes from 2 to 57 are sufficient.

Modifications by HackerRank

The test best * nextPrime >= limit might overflow.
The same result can be achieved this way:
best * nextPrime >= limit

best >= dfrac{limit}{nextPrime}

All variables are integers and thus rounding comes into play.
The correct formula is:
best >= dfrac{limit}{nextPrime} + dfrac{nextPrime - 1}{nextPrime}

best >= dfrac{limit + nextPrime - 1}{nextPrime}

My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
 
int main()
{
// enough primes for this problem
const unsigned int primes[] = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 57 };
 
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned long long limit;
std::cin >> limit;
 
// multiply until we hit our limit
unsigned long long bestPos = 1;
for (auto p : primes)
{
// continue until bestPos reaches or exceeds our input value
//__int128 next = bestPos * p;
//if (next >= limit)
// break;
// same code as before but more portable:
if (bestPos >= (limit + p - 1) / p)
break;
 
bestPos *= p;
}
 
std::cout << bestPos << std::endl;
}
return 0;
}

This solution contains 5 empty lines, 7 comments and 1 preprocessor command.

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 "1 10" | ./69

Output:

(please click 'Go !')

(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.
(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 1, 2017 submitted solution
April 26, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler069

My code solves 8 out of 8 test cases (score: 100%)

Difficulty

Project Euler ranks this problem at 10% (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=69 - 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-69-find-the-value-of-n-≤-1000000-for-which-nφn-is-a-maximum/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p069.java (written by Nayuki)
Go: github.com/frrad/project-euler/blob/master/golang/Problem069.go (written by Frederick Robinson)
Scala: github.com/samskivert/euler-scala/blob/master/Euler069.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.

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

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The 160 solved problems had an average difficulty of 21.8% at Project Euler and I scored 11,807 points (out of 13100) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
more about me can be found on my homepage.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !