Programmable gain non inverting amplifier
The standard way to achieve this is using a Voltage Controlled Amplifier. If you want uP control, you would typically use a D to A output to drive the VCA. 

Of course not all amplifiers are created equal, in terms of noise, THD, frequency range and so on, so some care is needed in selection of type. I link to the page of THAT Corp which specialises in VCA's for use in audio (where they are heavily used in compressors, limiters and other types of dynamic control).

By the way if you want an easy way to find standard (E24) resistor combinations for a particular gain (inverting or non-inverting) I wrote a book some years ago which tabulates that kind of thing.