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5

(Answering 1. only:) You have showed that the numbers $A_n$ cannot be rational. But Mills' constant is the limit of that sequence of numbers, not one of the numbers themselves. There is nothing to prevent a sequence of irrational numbers from having a rational limit. For a simpler example, consider the sequence $c_n=2^{1/n}$, with $\lim_{n \to \infty} c_n=...


22

What? You mean the Sophomore's Dream? (Actually, the "dream" is that $\int_0^1 x^{-x} \,\mathrm{d}x = \sum_{n=1}^\infty n^{-n}$, but this is just two representations of your value.) Your value appears in the ISC, associated with that sum. This sequence of digits appears in the OEIS as A073009 (with various references, including to Bernoulli's proof that ...



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