Problem 133

Repunit nonfactors

A number consisting entirely of ones is called a repunit. We shall define R(`k`) to be a repunit of length `k`; for example, R(6) = 111111.

Let us consider repunits of the form R(10^{n}).

Although R(10), R(100), or R(1000) are not divisible by 17, R(10000) is divisible by 17. Yet there is no value of `n` for which R(10^{n}) will divide by 19. In fact, it is remarkable that 11, 17, 41, and 73 are the only four primes below one-hundred that can be a factor of R(10^{n}).

Find the sum of all the primes below one-hundred thousand that will never be a factor of R(10^{n}).

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