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$\ce{CH3NHCH(CH3)CH2CH3}$
Should this be named as 2-(N-methyl)butanamine or 2-(methylamino)butane?

In other words, do we name the compound as 'alkanamine' or treat amino group as a substituent of butane?

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It should be N-methyl-2-butanamine. Amine is treated as a functional group attached. So we write it as alkanamine as it is the only functional group here.

enter image description here

Image source: ChemSpider

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    $\begingroup$ Note that the preferred IUPAC name for the compound drawn is in fact slightly different from what you have written: it should be N-methylbutan-2-amine. In particular the locant '2' should be placed immediately before the suffix 'amine', and the locant 'N' should be italicised. $\endgroup$ – orthocresol Mar 9 '18 at 15:10

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