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## Factorial — \(n!\)

The factorial of a positive integer \(n\), denoted by \(n!\), is the product of all positive integers less than or equal to \(n\):

$$n! = 1 × 2 × 3 × … × (n-2) × (n-1) × n$$

According to the convention for an empty product, \(0! = 1\).

For example,

$$5! = 1 × 2 × 3 × 4 × 5 = 120$$

\(\qquad\)

0! = 1

1! = 1

2! = 2

3! = 6

4! = 24

5! = 120

6! = 720

7! = 5040

8! = 40320

9! = 362880

10! = 3628800

\(\qquad\)

0! = 1.000000000000000000000000000

1! = 1.000000000000000000000000000

2! = 2.000000000000000000000000000

3! = 6.000000000000000000000000000

4! = 24.00000000000000000000000000

5! = 120.0000000000000000000000000

6! = 720.0000000000000000000000000

7! = 5040.000000000000000000000000

8! = 40320.00000000000000000000000

9! = 362880.0000000000000000000000

10! = 3628800.000000000000000000000

## Double Factorial — \(n!!\)

$$n!! = n × (n-2) × (n-4) × …$$

For example,

$$9!! = 9 × 7 × 5 × 3 × 1 = 945$$

for (x = 0, 10, print(x, “!! = “, f(x)));

\(\qquad\)

0!! = 1

1!! = 1

2!! = 2

3!! = 3

4!! = 8

5!! = 15

6!! = 48

7!! = 105

8!! = 384

9!! = 945

10!! = 3840

## Triple Factorial — \(n!!!\)

$$n!! = n × (n-3) × (n-6) × …$$

For example,

## Related to Factorials

### Primorial Numbers — \(n\#\)

The primorial (\(n\#\)) (sequence A002110 in the OEIS) is similar to the factorial, but with the product taken only over the prime numbers. The product of first n primes.

#### Primorial Numbers (First Definition)

OEIS A002110 Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.

For example,

$$11\# = 2 × 3 × 5 × 7 × 11 = 2310$$

for (x = 1, 10, print(prime(x), “# = “, f(x)));

\(\qquad\)

2# = 2

3# = 6

5# = 30

7# = 210

11# = 2310

13# = 30030

17# = 510510

19# = 9699690

23# = 223092870

29# = 6469693230

#### Primorial Numbers (Second Definition)

OEIS A034386 Primorial numbers (second definition): n# = product of primes <= n.

\\ f(n) = my(v = primes(primepi(n))); prod(i = 1, #v, v[i]);

f(n) = lcm(primes([2, n]))

for (x = 1, 15, print(prime(x), “# = “, f(x)));

\(\qquad\)

2# = 1

3# = 2

5# = 6

7# = 6

11# = 30

13# = 30

17# = 210

19# = 210

23# = 210

29# = 210

31# = 2310

37# = 2310

41# = 30030

43# = 30030

47# = 30030

### Superfactorial — sf(\(n\))

For example,

$$\textrm{sf}(4) = 1! × 2! × 3! × 4! = 288$$