Primes


R is a scalar or oneelement 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


L is an integer scalar whose meaning is as follows


R is a scalar consisting of a positive integer to which one of the above functions is applied.  
Z is an integer. 
Contents
Nth Prime Function
The Nth prime function (¯2πR) returns the R^{th} prime where 2 is the first prime. This function is sensitive to the index origin.
Previous Prime Function
The previous prime function (¯1πR) returns the prime that immediately precedes R.
Primality Test
The primality test function (0πR) returns a 1 if R is a prime and 0 if not.
Next Prime Function
The next prime function (1πR) returns the prime that immediately follows R.
Number Of Primes Function
The number of primes function (2πR) returns number of primes less than or equal to R.
Divisor Count Function
The divisor count function (10πR) 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 (11πR) returns the sum of the divisors of a number. It is the same as ×/(¯1+(∪f)*1+∪⍦f)÷¯1+∪f←πR^{1}. This function is used to recognize deficient, perfect, and abundant numbers.
Möbius Function
The Möbius function (12πR) 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. It is used in the Möbius Inversion Formula to invert general arithmetic functions.
Totient Function
The totient function (13πR) (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).