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This is a / question and so there isn't much context I can give.

Motivation:

Lately, I have been interested in the popular claim that the decimal expansion of $\pi$ contains every natural number. This property is not yet proven.

I asked the following question on MSE not too long ago: Numbers whose expansion contains every natural number (base $b$), but that are not (simply) normal.

Around the same time, I asked about the history of the idea on History of Science and Mathematics Stack Exchange: https://hsm.stackexchange.com/q/15491/6312

It occurred to myself and others whether there is a name for the property. Hence . . .

The Question:

Is there a name for numbers whose expansion base $b$ contains every natural number base $b$?

For full generality, fix $b$.

Context:

I'm aware of normal numbers.

The last time I did a number theory course was in 2015.

The property seems popular enough to warrant a name, and I'm confident it has been studied before in the context of normal numbers.

Please help :)

Shaun
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  • Did you look at the Mathematics Stack Exchange (MSE) answer I cited in my comment to your History of Science and Mathematics Stack Exchange question? I gave several names that have been used for this notion at the beginning, and the 19 February 2003 sci.math post that I cite in that MSE answer discusses how much larger this set is than the set of normal numbers -- most numbers ARE normal in the sense of Lebesgue measure and most numbers are NOT normal in the sense of Baire category. However, most numbers are disjunctive (to every base) both for Lebesgue measure and Baire category. – Dave L. Renfro Jun 17 '23 at 06:42
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    In addition to the various references one can obtain from the MSE answer I cited (previous comment), see the Wiki page for Disjunctive numbers. Also, for those interested, not only are most real numbers NOT normal in the sense of Baire category (indeed, almost ALL real numbers are not normal in the sense of Baire category), but almost ALL real numbers are not normal in VERY STRONG ways -- see the papers (continued) – Dave L. Renfro Jun 17 '23 at 06:57
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    by Olsen and Stylianou. For a discussion of some VERY STRONG ways to not be normal and yet these ways hold for almost all real numbers in the sense of Baire category, see the discussion that begins with "In this last section I'd like to discuss some extreme forms of non-normality that do not seem to be very well known" in this 18 September 2009 sci.math post (also cited in my earlier-mentioned MSE answer, by the way). – Dave L. Renfro Jun 17 '23 at 06:58
  • Thank you, @DaveL.Renfro. I invite you to type up an answer! I will accept it! – Shaun Jun 17 '23 at 07:04

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