Sequence of numbers whose factorial on prime factorisation contains prime powers of prime numbers, whose power is greater than $1$ or contains multiplicity of one for all prime numbers less than equal to number.
Sequence: $1, 2, 3, 4, 5, 8, 14, \ldots$
If $k$(prime) is to be checked for its presence in sequence, then if $k-1$ is in sequence, then $k$ is also in sequence.
$a(3)=4$ as $4!=(2^3)\cdot(3^1)$. Since multiplicity of $2$ is $3$, which is prime, and multiplicity of $3$ is $1$, so $4$ finds it's presence in the sequence.