What is the year of a Remington 22 short long octagon barrel rifle the last patent date on it is April161912 barrel is stamped 310340 and what is the value of them in good working order?

Sounds like you have a Remington Model 12 pump action rifle. Value would depend on over-all condition. How much finish (bluing/stock) remains and whether the bore is in shootable condition. If the bore is rusty and the rifling is pitted beyond use value drops considerably. Depending on condition, value could be from $200.00 to $450.00.