Scientific names to piano key numbers
The frequencies are on an exponential scale.  You can use any note as your base value.  On a Chromatic scale, each note is a twelfth step of an octave (doubling of frequency, same tone).  So, a very common formula is to use the 440Hz A note.  (Search on 432Hz to see a controversy about this.)

$$ f = 440 \cdot 2^{\frac{n}{12}} $$

Where $n$ is the semi-tone count from your base.

Here is a good reference:  Note names, MIDI numbers and frequencies





Upon rereading, I didn't really answer your question.  Because of the gaps in note names (black keys on the piano), this is more of a programming exercise than a simple formula.

In Python:


#=======================================================
def NoteConverter( ArgNoteName ):

        theLetter = ArgNoteName[0:1]
        theNumber = int( ArgNoteName[1:] )

        theOrdinal = "C D EF G A B".find( theLetter )

        theMidiNumber = theNumber * 12 + theOrdinal + 12

        return theMidiNumber

#=======================================================


This is for Midi numbering.  Subtract 20 to get your piano key number (on most pianos).