⍭ — Prime

Keys: Alt-Backslash. Ranks: 0 monadic, 0 0 dyadic

⍭N: the prime with N primes before it. ⍭0 is 2. Integral inputs; exact integer results, except that the primality test 1⍭ gives Booleans.

⍭⁻¹N: the number of primes below N, which is the index of a prime N.

⍭⍳8               ⍝ [2 3 5 7 11 13 17 19]ₓ
⍭⁻¹⍭⍳4            ⍝ [0 1 2 3]ₓ
⍭⁻¹1 2 3 4 5 6     ⍝ [0 0 1 2 2 3]ₓ

K⍭N selects an operation by the code K. The operations are like J’s p:, with different codes. Factor gives factors and factor tables.

K Result
¯2 Previous prime, strictly below N
1 N is prime
2 Next prime, strictly above N
3 Euler’s totient 
1⍭¯1 0 1 2 3 4     ⍝ $f $f $f $t $t $f
2⍭1 2 3 4 5        ⍝ [2 3 5 5 7]ₓ
¯2⍭3 4 5 6         ⍝ [2 3 3 5]ₓ
3⍭1 2 3 4 5 6      ⍝ [1 1 2 2 4 2]ₓ

Euler’s totient of N is N times the product of 1-÷p over the distinct prime factors p of N. For one exact N, 3⍭N gives the same result as N××/1ₓ-÷∪⨸N. The ₓ on 1 keeps the result exact.

N←700ₓ ⋄ N××/1ₓ-÷∪⨸N     ⍝ 240ₓ
3⍭700                    ⍝ 240ₓ

Unit cells; results assemble with fill. Primality testing is deterministic through 64 bits, probabilistic above that (false-positive bound 2⁻⁶⁴).

Errors

  • DOMAIN: invalid selector/input; previous prime at/below 2