Feedback factor of Armstrong Oscillator
is the feedback fraction equal to the square root of the ratio of the
  tickler inductance to the primary inductance?


Yes, that is the correct formula. Transformer inductance-ratio is the turns-ratio squared and this assumes perfect magnetic coupling. I think it's a fair assumption for two coils closely wound.

Background - the primary voltage sets up a magnetic flux that is shared between the two coils and the formula of note is Faraday's law of induction: -

\$V=-N\dfrac{d\phi}{dt}\$

Because the flux is shared by both coils, you can see that the input-output voltage ratio is also the ratio of the turns.

This means that the gain of the transistor circuit has to be bigger than the step-down factor introduced by the transformer's turns ratio. If the two inductors have a primary:secondary ratio of 9:1 then the number of turns is \$\sqrt{9:1}\$ = 3:1 and therefore the transistor circuit must have a gain greater than 3 to begin oscillations.