Appropriate tool for calculating limits of sequences over integers is DiscreteLimit.
Nevertheless it cannot compute our task. The limit does not exist. We can find it with calculating limes superior and limes inferior:
MaxLimit[1/(a Sin[Pi^6 a]), a -> ∞]
MinLimit[1/(a Sin[Pi^6 a]), a -> ∞]
∞ -∞
For an insight let's plot appropriate sequence
DiscretePlot[1/(a Sin[Pi^6 a]), {a, 1000, 1200}]
It is clearly seen that Sin[ Pi a] takes values between -1 and 1, however Sin[Pi^6 a] can approach -1, 0 and 1 with a very good approximation.