Find the primorial of numbers - JavaScript

The primorial of a number n is equal to the product of the first n prime numbers. This is different from factorial, which multiplies consecutive integers.

For example, if n = 4, we need the first 4 prime numbers: 2, 3, 5, and 7.

2 × 3 × 5 × 7 = 210

Let's write a JavaScript function that calculates the primorial of a given number.

Understanding Prime Numbers

First, we need a helper function to check if a number is prime:

const isPrime = n => {
    if (n === 1) {
        return false;
    } else if (n === 2) {
        return true;
    } else {
        for (let x = 2; x < n; x++) {
            if (n % x === 0) {
                return false;
            }
        }
        return true;
    }
};

// Test the isPrime function
console.log("First few prime numbers:");
for (let i = 1; i <= 10; i++) {
    if (isPrime(i)) {
        console.log(i + " is prime");
    }
}

First few prime numbers:
2 is prime
3 is prime
5 is prime
7 is prime

Complete Primorial Function

Now we can implement the primorial function that finds the first n prime numbers and calculates their product:

const isPrime = n => {
    if (n === 1) {
        return false;
    } else if (n === 2) {
        return true;
    } else {
        for (let x = 2; x < n; x++) {
            if (n % x === 0) {
                return false;
            }
        }
        return true;
    }
};

const primorial = num => {
    if (num === 0) {
        return 1; // By definition, primorial(0) = 1
    }
    
    let count = 1; // We start with first prime (2)
    let candidate = 3; // Next number to check
    let product = 2; // Start with first prime
    
    while (count < num) {
        if (isPrime(candidate)) {
            product *= candidate;
            count++;
        }
        candidate++;
    }
    
    return product;
};

// Test with different values
console.log("Primorial of 1:", primorial(1));
console.log("Primorial of 4:", primorial(4));
console.log("Primorial of 5:", primorial(5));
console.log("Primorial of 6:", primorial(6));

Primorial of 1: 2
Primorial of 4: 210
Primorial of 5: 2310
Primorial of 6: 30030

How It Works

The algorithm works in these steps:

1. Handle special case: if n = 0, return 1
2. Start with the first prime number (2) and set product = 2
3. Use a counter to track how many primes we've found
4. Check each subsequent number for primality
5. When we find a prime, multiply it to our product and increment counter
6. Continue until we've found n prime numbers

Alternative Implementation with Array

Here's another approach that first generates an array of prime numbers:

const generatePrimes = count => {
    const primes = [];
    let candidate = 2;
    
    while (primes.length < count) {
        let isPrime = true;
        for (let i = 2; i * i <= candidate; i++) {
            if (candidate % i === 0) {
                isPrime = false;
                break;
            }
        }
        if (isPrime) {
            primes.push(candidate);
        }
        candidate++;
    }
    return primes;
};

const primorialArray = num => {
    if (num === 0) return 1;
    
    const primes = generatePrimes(num);
    console.log(`First ${num} primes:`, primes);
    
    return primes.reduce((product, prime) => product * prime, 1);
};

console.log("Primorial of 4:", primorialArray(4));

First 4 primes: [ 2, 3, 5, 7 ]
Primorial of 4: 210

Comparison of Approaches

Conclusion

Primorial calculates the product of the first n prime numbers. The direct calculation approach is memory-efficient, while the array-based method offers better readability and debugging capabilities.