In 1876, Edouard Lucas showed that if an integer b exists such that b^(n-1)≡1 (mod n) and b^((n-1)/q)≢1 (mod n) for all prime divisors q of n-1, then n is prime, a result known as Lucas’s converse of Fermat’s little theorem. This result was considerably improved by Henry Pocklington in 1914 when he showed that it’s not necessary to know all the prime factors of n-1 to determine the primality of n. In this paper we optimize Pocklington’s primality test for integers of the form ap^k+1 where p is prime, a<4(p+1), k≥1. Precisely, this paper shows that if an integer b exists such that b^(n-1)≡1 (mod n) and n∤b^((n-1)/p)-1, then n is prime as opposed to Pocklington’s primality test that imposes the more stringent hypothesis that n and b^((n-1)/p)-1 be relatively prime. We also present a conjecture whose proof will significantly reduce the computations required to determine the primality of these integers.