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Not computable Non-computable but easily described arithmetical functions

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I have read about the existence of functions of the kind described in the title in several places, but never seen an instance of them. Sorry if this is too much an elementary question to be posted here.

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Not computable easily described arithmetical functions

I have read about the existence of functions of the kind described in the title, but never seen an instance of them. Sorry if this is too much an elementary question to be posted here.