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# 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:

This is equivalent to`echo "1 10" | ./69`

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.

(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 solved **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 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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