<< problem 178 - Step Numbers | Investigating in how many ways objects of two ... - problem 181 >> |
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.
My 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 I 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
).
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 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)
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.
The code contains #ifdef
s to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL
to produce the result for the original problem (default setting for most problems).
#include <iostream>
#include <vector>
#define ORIGINAL
int main()
{
// almost like a reverse prime sieve ...
unsigned int limit = 10000000;
#ifdef ORIGINAL
std::cin >> limit;
#endif
// count divisors of the number immediately after "limit", too
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]++;
#ifdef ORIGINAL
unsigned int result = 0;
for (unsigned int i = 2; i < limit; i++)
if (divisors[i] == divisors[i + 1])
result++;
std::cout << result << std::endl;
#else
// [index] => [numbers up to index which match the "neighbor" condition]
std::vector<unsigned int> count(limit + 1, 0);
// count numbers whose bigger neighbors has the same number of divisors
for (unsigned int i = 2; i < limit; i++)
{
count[i] = count[i - 1];
if (divisors[i] == divisors[i + 1])
count[i]++;
}
// simple lookup of results
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
unsigned int index;
std::cin >> index;
std::cout << count[index - 1] << std::endl;
}
#endif
return 0;
}
This solution contains 8 empty lines, 8 comments and 8 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.8 seconds on an 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=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
May 16, 2017 submitted solution
May 16, 2017 added comments
June 26, 2017 solve Hackerrank problem, too
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler179
My code solves 11 out of 11 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 25% (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.
Links
projecteuler.net/thread=179 - the best forum on the subject (note: you have to submit the correct solution first)
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 | |
the flashing problem is the one I solved most recently |
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I scored 13,486 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 178 - Step Numbers | Investigating in how many ways objects of two ... - problem 181 >> |