<< problem 207 - Integer partition equations | Flea Circus - problem 213 >> |

# Problem 211: Divisor Square Sum

(see projecteuler.net/problem=211)

For a positive integer n, let \sigma(n) be the sum of the squares of its divisors. For example,

\sigma^2(10) = 1 + 4 + 25 + 100 = 130.

Find the sum of all n, 0 < n < 64,000,000 such that \sigma^2(n) is a perfect square.

# My Algorithm

My approach works like a sieve:

- allocate enough memory for 64000000 numbers, each 64 bit (=> 512 MByte)

- fill it with zeros

- for each number 0 < i < 64000000: add i*i to each cell that is a multiple of i

- when done, check each cell whether it is a perfect square

The initial solution was small and produced the correct result in about 10 seconds.

However, its memory consumption was much, much higher than any of my other solutions.

Therefore I decided to move the algorithm to a new function `processSlice`

which works a bit smarter:

instead of processing everything at once, it only looks at all numbers [`from, to`

].

Reducing `to - from`

to about 4 million doesn't slow down the algorithm at all.

But I wanted to keep it below 20 MByte (for no good reason ...) and chose a `sliceSize = 2000000`

which takes about 14 seconds (40% slower).

## Alternative Approaches

You can play around with prime factorization. This should be a bit faster at the cost of probably doubling the code size.

# 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 <cmath>
// determine the sum of all numbers between "from" and "to" (inclusive both) which match the problem statement

unsigned long long processSlice(unsigned int from, unsigned int to)
{
std::vector<unsigned long long> sumSquares(to - from + 1, 0);
// like a prime sieve: add square of all divisors
for (unsigned long long i = 1; i <= to; i++)
{
// position of smallest multiple of i >= from
auto pos = (from / i) * i;
if (pos < from)
pos += i;
// add i^2 to all multiples of i
for (; pos <= to; pos += i)
sumSquares[pos - from] += i*i;
}
// find all sums that are perfect squares
unsigned int sum = 0;
for (size_t i = 0; i < sumSquares.size(); i++)
{
auto number = i + from;
auto current = sumSquares[i];
// compute integer square root
unsigned long long root = sqrt(current);
// iff root^2 = current then it's a perfect square
if (root * root == current)
sum += number;
}
return sum;
}
int main()
{
unsigned int limit = 64000000;
std::cin >> limit;
// how many number should be analyzed at once (=> influences memory consumption)
unsigned int sliceSize = 2000000;
// total sum
unsigned int sum = 0;
// start of current slice
unsigned int from = 1;
while (from < limit)
{
// end of current slice
auto to = from + sliceSize - 1;
if (to >= limit)
to = limit;
// process current slice
sum += processSlice(from, to);
// next slice
from = to + 1;
}
// print result
std::cout << sum << std::endl;
return 0;
}

This solution contains 10 empty lines, 14 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 100 | ./211`

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 15 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

Peak memory usage was about 18 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

May 31, 2017 submitted solution

May 31, 2017 added comments

# Difficulty

Project Euler ranks this problem at **50%** (out of 100%).

# Links

projecteuler.net/thread=211 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p211.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.

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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 207 - Integer partition equations | Flea Circus - problem 213 >> |