Is pentamine or pentylamine the correct name?
The following IUPAC guidelines suggest the correct systematic nomenclature for simple primary amines.

P-62.2.1.2
Primary amines are systematically named in the following ways:
(1) by adding the suffix ‘amine’ to the name of the parent hydride;
(2) by adding the name of the substituent group R - to the parent hydride ‘azane;
(3) by adding the name of the substituent group R - to the term ‘amine’ used as a preselected parent hydride name for NH3; this method is used only with monoamines.

Method (1) gives the preferred IUPAC name (PIN). So in your example this would give pentan-1-amine.
Method (3) gives what is often the more commonly used name, in your example this would give pentylamine.
Both names are 'fine' and in reality method (3) is the one that you will most often see used/written on bottles of reagent etc. But for the correct IUPAC identifier method (1) should be used.
You should also note that pentamine is a wholly different compound:

Other example:

By method (1). Methanamine
By method (2). Methylamine