Accepted answer
There is no count, and 31G is why: coring is a single event, not an accumulation. A needle cores when its bevel punches a full disc of rubber instead of parting it, and that either happens on a given puncture or does not. At 31G the bore is narrow enough that any disc it cuts is small and easy to miss under inspection. Technique moves the odds far more than the count does: enter at an angle, rotate the bevel, then bring it upright — that parts the septum rather than punching it. Inspect against a dark background before every draw, because the number of punctures a stopper survives is a property of that stopper and this is the only way to find it out.
Answer first: use the largest bore you tolerate for drawing and the smallest for injecting, because the two operations have opposite requirements.
Stopper coring — punching a disc of rubber into the solution — is a large-bore phenomenon. An 18G or 21G needle inserted straight and fast is the classic way to do it; inserting at a slight angle with the bevel up reduces the risk.
Fixed-needle insulin syringes are supplied in 29G to 31G and cannot be swapped for drawing, which is the trade-off against their much lower dead space.
The Hagen–Poiseuille relation gives flow proportional to the fourth power of radius, which is the quantitative basis for every gauge recommendation here.
Higher gauge is not automatically better; it is thinner, which has costs as well as benefits.
Flow goes as the fourth power of radius. That is why the difference feels so large.