Consider a small mass m connected to a larger mass M by a string. Now you
support the small mass m and let M hang below. When you release your hold,
m will accelerate at a = (M + m)/(mg). Just a re=wording of the previous a
= total weight/(mass of held portion). Note a spring is not essential - a
string will do to emphasize this phenomenon.
For the slinky I think the denominator depends on just how the thing is
held/released and on how much of the first turn is able to "move as a
whole", without bending - under the given circumstances.