<< problem 123 - Prime square remainders | Palindromic sums - problem 125 >> |

# Problem 124: Ordered radicals

(see projecteuler.net/problem=124)

The radical of n, rad(n), is the product of the distinct prime factors of n.

For example, 504 = 23 * 32 * 7, so rad(504) = 2 * 3 * 7 = 42.

If we calculate rad(n) for 1 <= n <= 10, then sort them on rad(n), and sorting on n if the radical values are equal, we get:

Unsorted Sorted

nrad(n) nrad(n)k

11 111

22 222

33 423

42 824

55 335

66 936

77 557

82 668

93 779

1010 101010

Let E(k) be the kth element in the sorted n column; for example, E(4) = 8 and E(6) = 9.

If rad(n) is sorted for 1 <= n <= 100000, find E(10000).

# My Algorithm

Similar to a prime sieve I find all multiples of each prime.

There is one struct `Radical`

for every number `n`

, its `product`

starts with `1`

and is multiplied by a prime factors of `n`

.

A simple comparison `operator<()`

will be used to sort all radicals.

A full sort isn't needed - there is the nice `nth_element`

algorithm in C++'s STL which ensures that a single element is at the correct position -

without a strict guarantee that other elements are perfectly sorted, too. This partial sort is often much faster than a full sort.

## Modifications by HackerRank

I don't properly re-use radicals for consecutive test cases and can't handle huge search spaces. I couldn't care less ...

# 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>
#include <algorithm>
// store a single radical

struct Radical
{
// current number
unsigned int n;
// product of all prime factors
unsigned int product;
// compare two object, prefer lower product (and lower n if products are equal)
bool operator<(const Radical& other) const
{
if (product != other.product)
return product < other.product;
return n < other.n;
}
};
// all relevant redicals

std::vector<Radical> rads;
// return a certain radical (1-based index)

unsigned int getNth(unsigned int index)
{
index--; // 1-based instead of 0-based
// partial sort
std::nth_element(rads.begin(), rads.begin() + index, rads.end());
return rads[index].n;
}
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int limit = 100000;
std::cin >> limit;
// initialize all radicals
rads.clear();
rads.reserve(limit + 1);
for (unsigned int i = 0; i <= limit; i++)
{
Radical current;
current.n = i;
current.product = 1;
rads.push_back(current);
}
// some big value to push zero to the back
rads[0].product = 999999999;
// compute radicals using a "sieve"
for (auto current : rads)
{
// process only primes
if (current.product != 1)
continue;
// adjust all of their multiples
for (unsigned int j = current.n; j <= limit; j += current.n)
rads[j].product *= current.n;
}
// get n-th element
unsigned int pos = 10000;
std::cin >> pos;
std::cout << getNth(pos) << std::endl;
}
return 0;
}

This solution contains 10 empty lines, 13 comments and 3 preprocessor commands.

# 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 "2 10 4 10 6" | ./124`

Output:

*Note:* the original problem's input `100000 10000`

__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 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

May 19, 2017 submitted solution

May 22, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler124

My code solves **4** out of **10** test cases (score: **30%**)

I failed **4** test cases due to wrong answers and **2** because of timeouts

# Difficulty

Project Euler ranks this problem at **25%** (out of 100%).

Hackerrank describes this problem as **advanced**.

*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=124 - **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-124-sorted-radical-function/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p124.java (written by Nayuki)

# 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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

# 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 123 - Prime square remainders | Palindromic sums - problem 125 >> |