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# Problem 179: Consecutive positive divisors

(see projecteuler.net/problem=179)

Find the number of integers 1 < n < 10^7, for which n and n + 1 have the same number of positive divisors.

For example, 14 has the positive divisors 1, 2, 7, 14 while 15 has 1, 3, 5, 15.

# Algorithm

Finally a simple problem ... I create an array `divisors`

with 10^7 entries.

Two nested loops go through all pairs (i,k) where i * k < 10^7 and increment each entry at `divisors[i * k]`

(in my code `j = i * k`

).

A second pass counts how often `divisors[n] == divisors[n + 1]`

.

## Note

8648640 has the most divisors: 447.

A `short`

uses less memory than an `int`

which caused less memory stalls (while still being able to store that maximum value of 447).

I saw a 20% performance boost on my system when switching from `int`

to `short`

.

Each number is divisible by 1 and by itself. When excluding those two divisors will still find the correct solution.

However, the program didn't become faster when starting the outer loop at `2`

(instead of `1`

) and the inner loop at `2*i`

(instead of `i`

).

# My code

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

#include <iostream>
#include <vector>
int main()
{
// almost like a reverse prime sieve
unsigned int limit = 10000000;
std::cin >> limit;
// will have the number of divisors for each number
std::vector<unsigned short> divisors(limit, 0);
// all numbers which can be a divisor ...
for (unsigned int i = 1; i < limit / 2; i++)
// and all of their multiples
for (unsigned int j = i; j < limit; j += i)
divisors[j]++;
// count numbers whose bigger neighbors has the same number of divisors
unsigned int result = 0;
for (unsigned int i = 2; i < limit; i++)
if (divisors[i] == divisors[i + 1])
result++;
std::cout << result << std::endl;
return 0;
}

This solution contains 5 empty lines, 5 comments and 2 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 20 | ./179`

Output:

*Note:* the original problem's input `10000000`

__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 **0.78** seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

Peak memory usage was about 22 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 16, 2017 submitted solution

May 16, 2017 added comments

# Difficulty

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

# Links

projecteuler.net/thread=179 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

# 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.

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<< problem 174 - Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements | Semiprimes - problem 187 >> |