Solution:

We claim that none of these numbers can ever be prime.

We prove this claim by noticing that the $n$th number is $$\sum_{k=0}^{2n} 10^k + 10^n = \sum_{k=0}^{n} 10^k +$$ $$\sum_{k=n}^{2n} 10^k =\sum_{k=0}^{n} 10^k + \sum_{k=0}^{n} 10^k \cdot 10^n$$ $$= (10^n+1)(\sum_{k=0}^{n} 10^k).$$ This shows that the number can be written as the product of two numbers greater than $1,$ so there are no primes.

Thus, the answer is A .