Whether there is a prime in sequence $\{1,12,123,1234,12345,123456,1234567,12345678,123456789,12345678910, \cdots\}$

Question

UPD: To make it clearer, here is a statement:

For sequence $A = \{i\in\mathbb{N}_+\mid a_i=a_{i-1}\times 10^{\lfloor(\lg(10n))\rfloor} + n\}$, where $\lg n=\log_{10} n$, show whether there is a prime number in the sequence. If there isn't, please prove it.

I ran into the problem accidentally.

To be clear, you can check the OEIS sequence A007908.

Using computer, I found no prime number up to 500, that is, $\overline{1234\cdots 499500}$.

Any form of help would be appreciated. Thanks in advance.