$\mathrm{ans} = \mid{ n \mid l \le n \le r \land (\mathrm{isprime}(n) \lor \exists m \lt n. \exists k. n = m^k ) }\mid$. This is done by simple implementation.

#!/usr/bin/env python2
import gmpy2
import string
import itertools
from pwn import * # https://pypi.python.org/pypi/pwntools
import argparse
parser = argparse.ArgumentParser()
args = parser.parse_args()
context.log_level = 'debug'
p = remote(args.host, args.port)

# proof of work
p.recvuntil('ASIS needs proof of work to start the Math challenge.\n')
_, salt, _, prefix, _ = p.recvline().split('"')
prefix = prefix.rstrip('.')
log.info('salt: %s', salt)
log.info('prefix: %s', prefix)
letters = string.digits + string.ascii_uppercase + string.ascii_lowercase
try:
for a in letters:
for b in letters:
for c in letters:
for d in letters:
x = a + b + c + d
if hashlib.sha256(x + salt).hexdigest().startswith(prefix):
raise StopIteration
except StopIteration:
log.info('result: %s', x)
p.sendline(x)

def is_valid(n):
n = gmpy2.mpz(n)
if gmpy2.is_prime(n):
return True
if bin(n).count('1') == 1:
return True
for k in itertools.count(2):
root, rem = gmpy2.iroot_rem(n, k)
if root == 2:
return False
if rem == 0:
return True
while True:
p.recvuntil("What's the number of primes or perfect powers like n such that: ")
l, _, _, _, r = p.recvline().split()
l, r = int(l), int(r)
log.info('l: %d', l)
log.info('r: %d', r)
ans = 0
for n in range(l, r+1):
if is_valid(n):
ans += 1
log.info('result: %d', ans)
p.send(str(ans))

p.recvall()