<< problem 386 - Maximum length of an antichain Migrating ants - problem 393 >>

A Harshad or Niven number is a number that is divisible by the sum of its digits.
201 is a Harshad number because it is divisible by 3 (the sum of its digits.)
When we truncate the last digit from 201, we get 20, which is a Harshad number.
When we truncate the last digit from 20, we get 2, which is also a Harshad number.
Let's call a Harshad number that, while recursively truncating the last digit, always results in a Harshad number a right truncatable Harshad number.

Also:
201/3=67 which is prime.
Let's call a Harshad number that, when divided by the sum of its digits, results in a prime a strong Harshad number.

Now take the number 2011 which is prime.
When we truncate the last digit from it we get 201, a strong Harshad number that is also right truncatable.
Let's call such primes strong, right truncatable Harshad primes.

You are given that the sum of the strong, right truncatable Harshad primes less than 10000 is 90619.

Find the sum of the strong, right truncatable Harshad primes less than 10^14.

# My Algorithm

A large portion of my code consists of the Miller-Rabin prime test and digitSum from my toolbox.

All single-digit numbers are Harshad numbers.
I generate all Harshad numbers with n digits by appending each possible digit 0 to 9 to all Harshad numbers with n-1 digits.
All strong Harshad numbers are stored in strong. If the n-1 digit number (I call it the "parent") is strong and the current number is prime, then I add it to the total sum.

# 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 4 | ./387

Output:

Note: the original problem's input 14 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 <algorithm>

// ---------- mulmod, powmod and Miller-Rabin test from my toolbox ----------

// return (a*b) % modulo
unsigned long long mulmod(unsigned long long a, unsigned long long b, unsigned long long modulo)
{
// (a * b) % modulo = (a % modulo) * (b % modulo) % modulo
a %= modulo;
b %= modulo;

// fast path
if (a <= 0xFFFFFFF && b <= 0xFFFFFFF)
return (a * b) % modulo;

// we might encounter overflows (slow path)
// the number of loops depends on b, therefore try to minimize b
if (b > a)
std::swap(a, b);

// bitwise multiplication
unsigned long long result = 0;
while (a > 0 && b > 0)
{
// b is odd ? a*b = a + a*(b-1)
if (b & 1)
{
result += a;
if (result >= modulo)
result -= modulo;
// skip b-- because the bit-shift at the end will remove the lowest bit anyway
}

// b is even ? a*b = (2*a)*(b/2)
a <<= 1;
if (a >= modulo)
a -= modulo;

// next bit
b >>= 1;
}

return result;
}

// return (base^exponent) % modulo => simple implementation
unsigned long long powmod(unsigned long long base, unsigned long long exponent, unsigned long long modulo)
{
unsigned long long result = 1;
while (exponent > 0)
{
// fast exponentation:
// odd exponent ? a^b = a*a^(b-1)
if (exponent & 1)
result = mulmod(result, base, modulo);

// even exponent ? a^b = (a*a)^(b/2)
base = mulmod(base, base, modulo);
exponent >>= 1;
}
return result;
}

// Miller-Rabin-test
bool isPrime(unsigned long long p)
{
// IMPORTANT: requires mulmod(a, b, modulo) and powmod(base, exponent, modulo)

// some code from             https://ronzii.wordpress.com/2012/03/04/miller-rabin-primality-test/
// with optimizations from    http://ceur-ws.org/Vol-1326/020-Forisek.pdf
// good bases can be found at http://miller-rabin.appspot.com/

// trivial cases
const unsigned int bitmaskPrimes2to31 = (1 <<  2) | (1 <<  3) | (1 <<  5) | (1 <<  7) |
(1 << 11) | (1 << 13) | (1 << 17) | (1 << 19) |
(1 << 23) | (1 << 29); // = 0x208A28Ac
if (p < 31)
return (bitmaskPrimes2to31 & (1 << p)) != 0;

if (p %  2 == 0 || p %  3 == 0 || p %  5 == 0 || p % 7 == 0 || // divisible by a small prime
p % 11 == 0 || p % 13 == 0 || p % 17 == 0)
return false;

if (p < 17 * 19) // we filtered all composite numbers < 17*19, all others below 17*19 must be prime
return true;

// fine-tuning for the problem:
// fast check of more small primes (up to 100)
if (p % 19 == 0 || p % 23 == 0 || p % 29 == 0 || p % 31 == 0 || p % 37 == 0 ||
p % 41 == 0 || p % 43 == 0 || p % 47 == 0 || p % 53 == 0 || p % 59 == 0 ||
p % 61 == 0 || p % 67 == 0 || p % 71 == 0 || p % 73 == 0 || p % 79 == 0 ||
p % 83 == 0 || p % 89 == 0 || p % 97 == 0)
return false;

// test p against those numbers ("witnesses")
// good bases can be found at http://miller-rabin.appspot.com/
const unsigned int STOP = 0;
const unsigned int TestAgainst1[] = { 377687, STOP };
const unsigned int TestAgainst2[] = { 31, 73, STOP };
const unsigned int TestAgainst3[] = { 2, 7, 61, STOP };
// first three sequences are good up to 2^32
const unsigned int TestAgainst4[] = { 2, 13, 23, 1662803, STOP };
const unsigned int TestAgainst7[] = { 2, 325, 9375, 28178, 450775, 9780504, 1795265022, STOP };

// good up to 2^64
const unsigned int* testAgainst = TestAgainst7;
// use less tests if feasible
if (p < 5329)
testAgainst = TestAgainst1;
else if (p < 9080191)
testAgainst = TestAgainst2;
else if (p < 4759123141ULL)
testAgainst = TestAgainst3;
else if (p < 1122004669633ULL)
testAgainst = TestAgainst4;

// find p - 1 = d * 2^j
auto d = p - 1;
d >>= 1;
unsigned int shift = 0;
while ((d & 1) == 0)
{
shift++;
d >>= 1;
}

// test p against all bases
do
{
auto x = powmod(*testAgainst++, d, p);
// is test^d % p == 1 or -1 ?
if (x == 1 || x == p - 1)
continue;

// now either prime or a strong pseudo-prime
// check test^(d*2^r) for 0 <= r < shift
bool maybePrime = false;
for (unsigned int r = 0; r < shift; r++)
{
// x = x^2 % p
// (initial x was test^d)
x = mulmod(x, x, p);
// x % p == 1 => not prime
if (x == 1)
return false;

// x % p == -1 => prime or an even stronger pseudo-prime
if (x == p - 1)
{
// next iteration
maybePrime = true;
break;
}
}

// not prime
if (!maybePrime)
return false;
} while (*testAgainst != STOP);

// prime
return true;
}

// ---------- now code specific to problem 387 ----------

// sum of all digits
unsigned int digitSum(unsigned long long x)
{
unsigned int result = 0;
while (x >= 10)
{
result += x % 10;
x      /= 10;
}
return result + x;
}

int main()
{
// up to 10^14
unsigned int maxDigits = 14;
std::cin >> maxDigits;

unsigned long long result = 0;

// all single-digit numbers are Harshad numbers
std::vector<unsigned long long> todo = { 1,2,3,4,5,6,7,8,9 };

std::vector<unsigned long long> strong;

// find all Harshad numbers with current number of digits
unsigned int numDigits = 1;
while (numDigits < maxDigits)
{
numDigits++;
std::vector<unsigned long long> next;

// process all Harshad numbers of the previous iteration
for (auto x : todo)
{
// "strong" is a sorted vector
bool isStrongParent = std::binary_search(strong.begin(), strong.end(), x);

// append every possible digit
for (unsigned int digit = 0; digit <= 9; digit++)
{
auto current = x * 10 + digit;
// prime and strong Harshad number of last digit is removed ?
if (isStrongParent && isPrime(current))
result += current;

// divisible by its digital sum ?
auto reduced = digitSum(current);
if (current % reduced == 0)
{
next.push_back(current);

// even a strong Harshad number ?
if (isPrime(current / reduced))
strong.push_back(current);
}
}
}

todo = std::move(next);
}

// and we're done
std::cout << result << std::endl;
return 0;
}


This solution contains 37 empty lines, 55 comments and 3 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in 0.02 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(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

August 2, 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 386 - Maximum length of an antichain Migrating ants - problem 393 >>
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