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There are this interesting properties about prime numbers, where a number of $n$ digits can have $k$ primes within it's digits

Examples: $1999$ has three primes in it, $\color{blue}{19}99, \color{blue}{199}9, \color{blue}{1999}$. Therefore we can say $1999$ is $4k_2$ prime (where $2$ here means the distinct digits of the number and $4$ is the number of primes within)

$23333$ contains five primes, $\color{blue}{2}3333, \color{blue}{23}333, \color{blue}{233}33, \color{blue}{2333}3, \color{blue}{23333}$. We can say $23333$ is $5k_2$ prime

$1327$ is not a $k$ prime, since although $13$ is a prime, $\color{blue}{132}7$ is not a prime (therefore, it doesn't have consecutive primes in it's digits)

Do we know of any conditions on $n,c$ where at-least a prime of form $ck_n$ exist?

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You might look at OEIS sequence A024770. This is not quite the same as what you're doing, though: for example $1999$ and $1327$ are not in A024770 since $1$ is not prime.

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