Harmonic Nodes and String Division

My daughter's fiancé is a formally trained musician, flutist, and composer. We were discussing harmonics last night as he was tuning my nephew's new ukulele, and he explained the minor seventh occurs when a string is divided into seven equal parts. He explained that this is purely coincidence. Do we have a chart that shows the interval by string subdivision? I had only thought about the 1/2=octave before, but I doubt that the pattern is random. Does it go something like this?

1/2=octave (e.g., G on the G string), at the 12th fret
1/3=fifth (e.g., D on the G string), in two places
1/4=octave (e.g., G on the G string), in two places not counting the twelve fret, which is a different octave
1/5=major third (e.g., B on the G string), in four places
1/6=fifth (e.g., D on the G string), in three places (in addition to where the lower D notes are)
1/7=minor seventh (e.g., F on the G string), in six places
1/8=octave, etc.
1/9=ninth?

 

This wiki page is consistent with this table.
http://en.wikipedia.org/wiki/Guitar_harmonics