<s>[INST] Prime numbers of the form $(1\times11\times111\times1111\times...)-(1+11+111+1111+...)$ Let
$$R(1) = 1-1,$$
$$R(2) = (1\times11) - (1+11),$$
$$R(3) = (1\times11\times111) - (1+11+111),$$ and so on...
$$R(4)=1355297\quad\text{(a prime number!)}$$
$R(4)$ is the only prime I found of such form up to $R(200)$. Are there anymore primes of such form?
 [/INST] I wanted to post this as a comment but it was too long.
I don't know if it will help but consider to rewrite your function in this way:
$$R(n) = (1 \times 11 \times 111 \times \ldots) - (1 + 11 + 111 + 1111 + \ldots)$$
$$R(n) = \left((10^0) \times (10^0 + 10^1) \times (10^0 + 10^1 + 10^2) \times \ldots\right) - \left((10^0) + (10^1 + 10^0) + (10^2 + 10^1 + 10^0) + \ldots \right)$$
Now with a bit of maths you can check that 
$$\left((10^0) \times (10^0 + 10^1) \times (10^0 + 10^1 + 10^2) \times \ldots\right) = \prod_{k = 1}^n \frac{10^k - 1}{9}$$
For what concerns the second part, it's more amusing. Indeed we have
$$\left((10^0) + (10^1 + 10^0) + (10^2 + 10^1 + 10^0) + \ldots \right)$$
But as we tend to $n$, we can easily check that there will be $n$ terms of $10^0$, $n-1$ terms of $10^1$, $n-2$ terms of $10^2$ and so on, which means
$$\left((10^0) + (10^1 + 10^0) + (10^2 + 10^1 + 10^0) + \ldots \right) = \sum_{k = 0}^{n-1} (n-k)10^k$$
Now with a bit of help (mathematica rules), the productory gives
$$\prod_{k = 1}^n \frac{10^k - 1}{9} = 9^{-k} {Q}_p (10, 10, k)$$
Where ${Q}_p (10, 10, k)$ is the so called q-Pochammer symbol, whose definition is
$${Q}_p (a, q, k) = \prod_{i = 0}^{k-1} (1 - aq^i)$$
Ref https://en.wikipedia.org/wiki/Q-Pochhammer_symbol
Whereas the sum gives
$$\sum_{k = 0}^{n-1} (n-k)10^k = \frac{1}{81}\left(10^{1+n} - 9n - 10\right)$$
Thus your function is
$$R(n) = 9^{-k} {Q}_p (10, 10, k) - \frac{1}{81}\left(10^{1+n} - 9n - 10\right)$$
As I said, I don't know if this helps, but it's a good way to check $R(n)$ quite fast.
Please If I made some mistake, make a comment. I liked this question and I immediately got involved in trying to find a suitable general form for $R(n)$.
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