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Problem 179: Consecutive positive divisors

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:

Input data (separated by spaces or newlines):
Note: Enter the upper search limit

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

Difficulty

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

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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The 133 solved problems had an average difficulty of 16.9% at Project Euler and I scored 11,174 points (out of 12300) at Hackerrank's Project Euler+.
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