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In the binary representation of $2^n$ I know there are $n-1$ zeros. Yet I'm not sure how can I find the function to calculate the number of zeros in the decimal representation. It seems to be possible using a function translating from binary to decimal but I can't manage to put the things together. Would be happy for any help.

Aladin
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  • I do not think that such a function exists (in the sense of a formula). You simply have to count the zeros in general. – Peter Sep 21 '19 at 17:51
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    This just OEIS A027870. There appears to be no closed form. – Peter Foreman Sep 21 '19 at 17:58
  • If so, theoretically, do you mean it is unknown and very hard to answer? Or it can be proved that there is no function (that can be expressed as a formula, say, using the symbols of addition, multiplication and power)? – Aladin Sep 21 '19 at 18:00
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    Studying representations of powers in various bases is a hard problem, see e.g. problems mentioned in this answer. It dare to guess it's completely open whether there always is at least a single zero in the decimal expansion of $2^n$ for large enough $n$ (though we would expect, on average, the digits to be more or less evenly distributed, so 1/10 of the digits "should" be zeros). Edit: Seems my guess was right, relevant OEIS – Wojowu Sep 21 '19 at 18:03

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