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Recall the definition of a normal number: a number $a$ is called normal to base $b$ if in its expansion in base $b$, the number of appearances of every single single string of $k$ base-$b$ digits in its first $n$ digits divided by $n$ tends to $\frac1{b^k}$ as $n$ tends to infinity. A number is called absolutely normal or just normal if it is normal to every base $b \geq 2$.

Given a fixed base $b$, there is a wealth of explicit examples of numbers normal to base $b$, such as a base $b$ Champernowne or Copeland-Erdős constant. There are some more elusive examples of absolutely normal numbers: any Chaitin's constant is normal, but such numbers are always uncomputable; apparently, a computable normal number is constructed in this article, although it does not actually calculate any of its digits.

Now, my question: are there numbers which are normal to one base, but not another? If so, can we explicitly give an example of such a number?

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There are infinitely many such numbers. One family of such numbers are the Stoneham constants, defined as

$$ \alpha_{b,c} = \sum_{n \ge 1}\frac{1}{c^n b^{c^n}} $$

where $\gcd(b,c) = 1$. Stoneham constants are known to be normal in base $b$ but not in base $B = b^p c^q r$ where $p,q,r \ge 1$ and neither $b$ nor $c$ divide $r$.

For an more details explicit construction of such numbers, you can refer to the paper by David H. Bailey and Jonathan M. Borwein.

http://www.davidhbailey.com/dhbpapers/nonnormality.pdf

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