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I was wondering what the shortest way to represent any given number is. For example, $387420489=9^9$. So, for this case, the smallest representation is of order 2 (2 numbers). Alternatively, $10=2\cdot5$ also has 2, but it began with 2, so there is nothing constructive there.

The symbols permitted in the expression are $+,-,\times$ and taking exponential is also permitted.

Edit: To make this into a question, is there a general form to numbers that have a least character representation that is less than their natural representation?

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    Although for any given number one can search, I don't see a way to answer the question in general. What are you looking for?2011-11-17
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    @Ross, is there any interest in this type of problem? I was looking for a general solution.2011-11-17
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    Just out of curiosity, why are you not allowing division?2011-11-17
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    One could certainly write a program to generate all the strings that involve a given number of digits and evaluate them, giving the list of numbers expressible by that many digits. I do not see a general solution to finding that $5686676=7^8-5^7$ for example2011-11-17
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    @Robert, I have no good reason. If that were allowed though, I would not know how to answer why no $!$ or something else more fun. Perhaps the $-$ was a bad idea.2011-11-17
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    What Ross has said. I note that the smallest (positive integer) number that has a representation shorter than just writing it in decimal is $125=5^3$, followed by $128=2^7$, $216=6^3$, $243=3^5$, $256=2^8=4^4$, $343=7^3$, $512=2^9=8^3$, $625=5^4$, $729=3^6=9^3$.2011-11-17
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    @Ross, I hope the edit helps.2011-11-18
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    The sequence of numbers @Gerry gave is in the [OEIS](http://oeis.org/A104233).2011-11-18
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    @J.M. Yes you are correct, but the sequence given in the OEIS is more restrictive as they allow a lot more parameters. Namely, they allow division and bracketing. However, that being said, I think finding a general form may be too tricky2011-11-18
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    To make this interesting you should allow division and bracketing, and only allow the digit 1. In that case I think that the expansion of the prime factorization may be the shortest expression.2013-02-15

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