<< problem 191 - Prize Strings | Prime triplets - problem 196 >> |
Problem 193: Squarefree Numbers
(see projecteuler.net/problem=193)
A positive integer n is called squarefree, if no square of a prime divides n, thus 1, 2, 3, 5, 6, 7, 10, 11 are squarefree, but not 4, 8, 9, 12.
How many squarefree numbers are there below 2^50?
My Algorithm
My program counts all non-squarefree numbers and then displays 2^50 - notSquarefree.
Any number can be written as a product of its prime factors:
x = {p_1}^{e_1} * {p_2}^{e_2} * ... * {p_n}^{e_n}
If x is squarefree, then all its exponents are e_1 = e_2 = ... = e_n = 1.
A non-squarefree number, let's call it "squary", has at least one exponent e_i > 1.
The largest prime factor to be considered for finding squaries must be {p_i}^2 < 2^50 or p_i < sqrt{2^50}.
The first step in my program is to count the number of distinct prime factors for all numbers below sqrt{2^50} (these numbers are my "base numbers").
The idea is that all multiples k * b^2 of these "base numbers" b aren't squarefree.
However, all relevant "base number" must be squarefree themselves. That's why I track all "squary base numbers" in my ignore
array.
The number of multiples k * b^2 of a "squary base number" i
is numMultiples = limit / (b * b)
.
There is one more problem - some numbers are counted multiple times.
For example 192 = 2^2 * 7^2 is a multiple of 2, 7 and 14 (=2*7).
A pretty sweet workaround solves this problem:
- if the number of prime factors of a base number is odd, then add the number of multiples
- if the number of prime factors of a base number is even, then the number of multiples was already added twice: once for each pair of prime factors
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 13 | ./193
Output:
Note: the original problem's input 1125899906842624
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 <vector>
#include <cmath>
int main()
{
unsigned long long limit = 1ULL << 50;
std::cin >> limit;
// largest prime factor that may appear as a square
unsigned long long root = (unsigned long long)sqrt(limit);
// count number of distinct prime factors
std::vector<unsigned char> numPrimeFactors(root, 0);
// does any prime factor occurs multiple times ?
std::vector<bool> ignore(root, false);
for (unsigned long long prime = 2; prime < root; prime++)
{
// skip if not a prime number
if (numPrimeFactors[prime] != 0)
continue;
// add the prime factor to all multiples
for (auto j = prime; j < root; j += prime)
numPrimeFactors[j]++;
// all multiples have at least one prime factor multiples times, mark as invalid
auto square = prime * prime;
for (auto j = square; j < root; j += square)
ignore[j] = true;
}
// count all numbers that are not squarefree
unsigned long long notSquarefree = 0;
for (unsigned long long base = 2; base < root; base++)
{
// at least one prime factor occurs multiple times ?
if (ignore[base])
continue;
// all multiples are not squarefree
auto square = base * base;
auto numMultiples = limit / square;
// if the number of prime factors is odd, then these multiples are new
if (numPrimeFactors[base] % 2 == 1)
notSquarefree += numMultiples;
else // else: when even number of prime factors, then we have seen these numbers before
notSquarefree -= numMultiples;
}
// display result
auto result = limit - notSquarefree;
std::cout << result << std::endl;
return 0;
}
This solution contains 9 empty lines, 11 comments and 3 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 39 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
June 15, 2017 submitted solution
June 15, 2017 added comments
Difficulty
Project Euler ranks this problem at 55% (out of 100%).
Links
projecteuler.net/thread=193 - 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:
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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 | |
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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 !!!
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