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(Python)
from sympy import isprime
from itertools import product
def sign(n): return (n > 0) - (n < 0)
def unds(n):
s = str(n)
if len(s) == 1: return True
signs = set(sign(int(s[i-1]) - int(s[i])) for i in range(1, len(s), 2))
if len(signs) > 1: return False
if len(s) % 2 == 0: return signs == {1} or signs == {-1}
return sign(int(s[-1]) - int(s[-2])) in signs - {0}
def candidate_pals(n): # of length 2n + 1
if n == 0: yield from [2, 3, 5, 7]; return # one-digit primes
for rightbutend in product("0123456789", repeat=n-1):
rightbutend = "".join(rightbutend)
for end in "1379": # multi-digit primes must end in 1, 3, 7, or 9
left = end + rightbutend[::-1]
for mid in "0123456789": yield int(left + mid + rightbutend + end)
def a(n): return sum(1 for p in candidate_pals(n) if unds(p) and isprime(p))
(Python)
from sympy import isprime
def f(w, dir):
if dir == 1:
for s in w:
for t in range(int(s[-1])+1, 10):
yield s+str(t)
else:
for s in w:
for t in range(0, int(s[-1])):
yield s+str(t)
c = 0
for d in '123456789':
x = d
for i in range(1, n+1):
x = f(x, (-1)**i)
c += sum(1 for p in x if isprime(int(p+p[-2::-1])))
if n > 0:
y = d
for i in range(1, n+1):
y = f(y, (-1)**(i+1))
c += sum(1 for p in y if isprime(int(p+p[-2::-1])))
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