p: 0
2
   p: i. 15
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

The inverse of p: is the number of primes less than the argument, often denoted by π(n):

   pi=: p:^:_1
   pi i. 15
0 0 0 1 2 2 3 3 4 4 4 4 5 5 6
   (] , pi ,: p:@pi) i.15
0 1 2 3 4 5 6 7  8  9 10 11 12 13 14
0 0 0 1 2 2 3 3  4  4  4  4  5  5  6
2 2 2 3 5 5 7 7 11 11 11 11 13 13 17

   y=: (2^31)-1
   ] a=: pi y
105097564
   ] b=: p: a
2147483647
   b=y
1

_4 p: y is the next prime smaller than y and is the same as
<:^:(0&p:)^:_"0 >.y-1 .

_1 p: y is the same as p:^:_1 y , the number of primes less than y .

0 p: y is the same as -.1 p: y .

1 p: y is 1 if and only if y is prime.

2 p: y is the same as __ q: y , a 2-row table of the prime factors and exponents in the factorization of y .

3 p: y is the same as q: y , the list of prime factors of y whose product is equal to y .

4 p: y is the next prime larger than y and is the same as
>:^:(0&p:)^:_"0 <.y+1 .

5 p: y computes the totient function (Euler’s phi function) of y , the number of non-negative integers less than y relatively prime to it, the sum +/1=y+.i.y .

Currently, arguments larger than 2^31 are tested to be prime according to a probabilistic algorithm (Miller-Rabin).