# Primes

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
 Z←πR returns an integer vector consisting of the prime factors of R.
R is a scalar or one-element vector consisting of a positive integer to be factored.
Z is an integer vector whose values are the prime factors of R.

For example,

π120
2 2 2 3 5

×/⎕←π⎕←¯1+2*62
4611686018427387903
3 715827883 2147483647
4611686018427387903

π¯1+2*61
2305843009213693951

⍴π1
0

 Z←LπR returns an integer scalar whose value depends upon which number-theoretic function is chosen by L to be applied to R.
L is an integer scalar whose meaning is as follows
 0 Divisor count function 1 Divisor sum function 2 Primality test function 3 Next prime function 4 Previous prime function 5 Nth prime function 6 Number of primes function 7 Möbius function function 8 Euler totient function
R is a scalar consisting of a positive integer to which one of the above functions is applied.
Z is an integer.

## Divisor Count Function

The divisor count function (0πR) returns the number of divisors of a number. It is the same as ×/1+∪⍦πR where πR returns the prime factors of R and ∪⍦ counts the number of occurrences of unique elements (in this case, the exponent vector of the unique primes). A divisor then consists of the product of zero or more of the unique primes which is why ×/1+ counts them.

## Divisor Sum Function

The divisor sum function (1πR) returns the sum of the divisors of a number. It is the same as ×/(¯1+(∪f)*1+∪⍦f)÷¯1+∪f←πR1. This function is used to recognize deficient, perfect, and abundant numbers.

## Primality Test

The primality test function (2πR) returns a 1 if R is a prime and 0 if not.

## Next Prime Function

The next prime function (3πR) returns the prime that immediately follows R.

## Previous Prime Function

The previous prime function (4πR) returns the prime that immediately precedes R.

## Nth Prime Function

The Nth prime function (5πR) returns the Rth prime where 2 is the first prime.

## Number Of Primes Function

The number of primes function (6πR) returns number of primes less than or equal to R.

## Möbius Function

The Möbius function (7πR) returns information about the square free properties of R. If R is square free, the function returns 1 if R has an even number of prime factors, and ¯1 if it has an odd number of prime factors. If the argument is not square free, the function returns 0.

## Totient Function

The totient function (8πR) (also called Euler's Totient Function) returns the number of positive integers less than or equal to R that are relatively prime to it (i.e., having no common positive factors other than 1).