Primes
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| 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
c←×/(a b)←1π?2⍴10*25x ⍝ Create two random 25-digit primes and their product
a b c
213379889007584782100623 7822072437371562999056237 1669072948495632279507879309030439547039369735651
πc ⍝ Factor the 50-digit number
213379889007584782100623 7822072437371562999056237
⎕T-(πc)⊢⊢⎕T ⍝ How many seconds of CPU time to factor a 50-digit number?
0.6204937084112316
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L is an integer scalar whose meaning is as follows
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| R is an array consisting of positive integers to which one of the above functions is applied, element by element. | ||||||||||||||||||
| Z is an integer array of the same shape as R. |
Rth Prime Function
The Rth prime function (¯2πR) returns the Rth prime where 2 is the first prime. This function is sensitive to the index origin.
For example, how many primes are less then or equal to 1000003?
¯2π1000003 15485927
Previous Prime Function
The previous prime function (¯1πR) returns the prime that immediately precedes R.
For example, what is the prime that immediately precedes 1000000?
¯1π1000000 999983
Primality Test
The primality test function (0πR) returns a 1 if R is a prime and 0 if not.
For example, is 1000003 a prime?
0π1000003 1
List the primes up to 100
⍸0π⍳100 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
Next Prime Function
The next prime function (1πR) returns the prime that immediately follows R.
For example, what is the next prime after 1000000?
1π1000000 1000003
Number Of Primes Function
The number of primes function (2πR) returns number of primes less than or equal to R.
For example, what is the 15485927th prime?
2π15485927 1000003
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←πR1. 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).
Examples
Add together the first 100 primes:
The Rth prime function (¯2πR) gives the value of the Rth prime, as in
¯2π100 541
The Index function (⍳R) produces a vector of integers of length R, as in
⍳¯2π100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... 541
The Primality Test function (0πR) returns a 1 if the corresponding element in R is a prime, 0 otherwise, as in
0π⍳¯2π100 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 ... 1
The Indices function (⍸R) converts the argument R to indices (equivalent to (,R)/⍳×/⍴R), as in
⍸0π⍳¯2π100 2 3 5 7 11 13 17 19 23 29 31 37 41 43 ... 541
Finally, those numbers may be added together using plus reduction (+/), as in
+/⍸0π⍳¯2π100 24133
