<< problem 20 - Factorial digit sum | Names scores - problem 22 >> |
Problem 21: Amicable numbers
(see projecteuler.net/problem=21)
Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
If d(a) = b and d(b) = a, where a!=b, then a and b are an amicable pair
and each of a and b are called amicable numbers.
For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore d(220) = 284.
The proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.
Evaluate the sum of all the amicable numbers under 10000.
My Algorithm
My function getSum
returns the sum of all proper divisors of x
.
The brute-force approach works but turns out to be too slow: a for
-loop from 2 to x-1
.
But we can easily reduce the number of iterations:
For each proper divisor i of x there is another divisor j=x/i, except when i^2=x.
More interesting, if we assume i<j then i<=sqrt{x}.
A simple loop running from 2 to sqrt{x} check whether x mod i == 0 (proper divisor)
and then adds i as well as j (if i!=j) to the result.
A precomputation step in main
finds the sums of all divisors of all numbers i
below 100000 and calls it sibling = getSum(i)
.
If getSum(sibling) = i
then i,sibling
is an amicable pair.
Those two numbers are stored in an std::set
named amicables
.
For each test case, the sum of all relevant numbers is printed.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1 300" | ./21
Output:
Note: the original problem's input 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)
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.
#include <iostream>
#include <set>
// generate sum of all divisor's of x (where x > 1)
unsigned int getSum(unsigned int x)
{
// find all factors:
// look only for the "smaller" divisors <= sqrt(x)
// and they have a "bigger" brother x / divisor
// 1 is always a divisor, but not the number x itself
unsigned int divisorSum = 1;
// therefore start at 2
for (unsigned int divisor = 2; divisor * divisor <= x; divisor++)
if (x % divisor == 0)
{
divisorSum += divisor;
// add the "bigger brother"
auto otherDivisor = x / divisor;
// except square numbers
if (otherDivisor != divisor)
divisorSum += otherDivisor;
}
return divisorSum;
}
int main()
{
// contain all numbers which are part of an amicable pair
std::set<unsigned int> amicables;
// precomputation step:
// find all amicable numbers <= 100000
const unsigned int MaxAmicable = 100000;
for (unsigned int i = 2; i <= MaxAmicable; i++)
{
auto sibling = getSum(i);
// found a pair ?
if (i == getSum(sibling) && i != sibling)
{
amicables.insert(i);
amicables.insert(sibling);
}
}
// and now start processing input
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int x;
std::cin >> x;
// just look up all suitables numbers
unsigned int sum = 0;
for (auto i : amicables)
{
// discard those that are too big
if (i > x)
break;
// note: an set::set is sorted ascendingly by default
// yes, accept that amicable number
sum += i;
}
std::cout << sum << std::endl;
}
return 0;
}
This solution contains 11 empty lines, 17 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.20 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
February 24, 2017 submitted solution
April 4, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler021
My code solves 4 out of 4 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 5% (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 rarely an option.
Similar problems at Project Euler
Problem 23: Non-abundant sums
Note: I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.
Links
projecteuler.net/thread=21 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
C# www.mathblog.dk/project-euler-21-sum-of-amicable-pairs/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/20-29/problem21.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p021.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/21 Amicable numbers.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem021.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p021.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p021.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler021.scala (written by Michael Bayne)
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 20 - Factorial digit sum | Names scores - problem 22 >> |