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I came across the following interesting objects. Self-locating strings within $\pi$: numbers $n$ such that the string $n$ is at position $n$ (after the decimal point) in decimal digits of $\pi$. The OEIS sequence https://oeis.org/A057680 lists the first $9$. My question is: is anything known about whether this sequence of numbers contains infinite terms? In other words, are there infinite self-locating strings within the decimal expansion of $\pi$?

I know that it is not known whether $\pi$ is a normal number. But even assuming it is, would this automatically imply that the number of self-locating strings is infinite? It seems to me that the layman assertion that 'every finite string would occur somewhere' in the expansion of a normal number does not necessarily imply that 'every finite string would occur at a prescribed position'. Am I wrong?

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    $\begingroup$ The problem for $\frac19$ is simpler :) $\endgroup$ – Hagen von Eitzen May 2 '16 at 15:04
  • $\begingroup$ You are right. Even assuming normality, we cannot deduce this stronger property. Which position counts ? The first or the last of the string ? $\endgroup$ – Peter May 2 '16 at 15:30

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