Which ukulele harmonic corresponds to which fret position?

Question

According to this page, ukulele natural harmonics can be played on the 12th (octave), 9th, 7th (octave+fifth), 5th (2octaves) and 3rd frets.

Which interval is obtained at the 9th and 3rd frets?

Answer 1

Harmonics can be played at any position where overtones have a "node", which is all integral divisions of the string. The 12th fret is at 1/2 of the string, the 7th and the 19th fret correspond to 1/3 and 2/3, The 5th fret and the 24th fret correspond to 1/4 and 3/4, slightly below 4th freth, 9th fret, 16th fret correspond to 1/5, 2/5, 3/5. 3rd corresponds to 1/6. Between 2/3, slightly under 6, slightly under 10, 15, 22 is 1/7, 2/7, 3/7, 4/7, 5/7. Slightly above 2nd fret, 8th fret, 16th fret correspond to 1/8, 3/8, 5/8. And so it goes on, but the higher you get the harder it becomes to pinpoint a node and the less energy is there, so it gets harder to get a nice harmonic.

1/2 then produces multiple of the first overtone, which is an octave. 1/3, 2/3 produce the second overtone, which is a fifth plus one octave. 1/4, 3/4 produce multiples of the third overtone, which is two octaves. 1/5, 2/5, 3/5, 4/5 produce the fourth overtone which is a (pure) major 3rd plus two octaves. 1/6, 5/6 produce the fifth overtone, which is a fifth plus two octaves, 1/7 - 6/7 produce the sixth overtone, which is a natural seventh. 1/8, 3/8, 5/8, 7/8 produce the seventh overtone, which is three octaves. And so on.

Answer 2

The 3rd fret is two octaves plus a fifth.
(see https://liveukulele.com/lessons/techniques/harmonics/#natural)

The 9th fret is two octaves plus a major third.
(see https://fretsource-guitar.weebly.com/natural-harmonics-map.html. Although the source regards guitar, it will be the same for ukulele.)

As a supplement, the Wikipedia page regarding the harmonic series may be of interest.