<< problem 351 - Hexagonal orchards Cyclic numbers - problem 358 >>

# Problem 357: Prime generating integers

Consider the divisors of 30: 1,2,3,5,6,10,15,30.
It can be seen that for every divisor d of 30, d+30/d is prime.

Find the sum of all positive integers n not exceeding 100 000 000
such that for every divisor d of n, d+n/d is prime.

# My Algorithm

Each number n has at least two divisors: 1 and n itself.
Therefore for each candidate n the following must be prime numbers:

• d + n/d = 1 + n/1 = 1 + n and
• d + n/d = n + n/n = n + 1 → which is the same as before
And indeed: for each divisor d there is some number e such that e = n / d (→ e * d = n).
e + n/e = n/d + n/(n/d) = n/d + d
I only have to check divisor d < e, hence d <= sqrt{n}.

In short, each candidate's successor must be a prime number.
For n=1 all conditions are fulfilled and it's a matching number.
For n>1 the successor can only be an odd prime number, therefore n>1 must be even.

And if n is even, then 2 is always a divisor:
that means that 2 + n/2 must be a prime number, too. However, 2 + n/2 can only be prime if n/2 is odd.
Now a stricter condition for n>1 is: n = 4k + 2n must be a multiple of 2, but not of 4.

## Note

My prime sieve was copied from my toolbox.

# 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):

This is equivalent to
echo 30 | ./357

Output:

Note: the original problem's input 100000000 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>

// odd prime numbers are marked as "true" in a bitvector
std::vector<bool> sieve;

// return true, if x is a prime number
bool isPrime(unsigned int x)
{
// handle even numbers
if ((x & 1) == 0)
return x == 2;

// lookup for odd numbers
return sieve[x >> 1];
}

// find all prime numbers from 2 to size
void fillSieve(unsigned int size)
{
// store only odd numbers
const unsigned int half = size >> 1;

// allocate memory
sieve.resize(half, true);
// 1 is not a prime number
sieve[0] = false;

// process all relevant prime factors
for (unsigned int i = 1; 2 * i*i < half; i++)
// do we have a prime factor ?
if (sieve[i])
{
// mark all its multiples as false
unsigned int current = 3 * i + 1;
while (current < half)
{
sieve[current] = false;
current += 2 * i + 1;
}
}
}

int main()
{
unsigned int limit = 100000000;
std::cin >> limit;

fillSieve(limit + limit/2);

// 1 is the only odd number where i+1 is prime
unsigned long long sum = 1;

// scan only even numbers >= 2 of the form 2+4k
for (unsigned int n = 2; n <= limit; n += 4)
{
// quick check for divisor = 1
if (!isPrime(1 + n))
continue;
// n is always divisible by 2, perform a second quick check for divisor = 2
if (!isPrime(2 + n/2))
continue;

// check all potential divisors 3 ... sqrt(n)
bool valid = true;
for (unsigned int divisor = 3; divisor*divisor <= n; divisor++)
{
// is it a divisor ?
if (n % divisor != 0)
continue;

// d + n/d
if (!isPrime(divisor + n / divisor))
{
valid = false;
break;
}

// other divisor is n/d:
// n/d + n/(n/d)
// = n/d + d => same as above
}

// found another value
if (valid)
sum += n;
}

// display result
std::cout << sum << std::endl;
return 0;
}


This solution contains 15 empty lines, 23 comments and 2 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in 1.6 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 11 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

July 8, 2017 submitted solution

# Difficulty

Project Euler ranks this problem at 10% (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 351 - Hexagonal orchards Cyclic numbers - problem 358 >>
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