<< problem 191 - Prize Strings Prime triplets - problem 196 >>

# Problem 193: Squarefree Numbers

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
→ in that case, subtract it

# 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 maximum number up to which the program should count all squarefree numbers

This is equivalent to
echo 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. Or just jump to my GitHub repository.

       #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

# Difficulty

Project Euler ranks this problem at 55% (out of 100%).

# 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 [new] the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.
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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 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.

 << problem 191 - Prize Strings Prime triplets - problem 196 >>
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