This note will establish formulae for the asymptotic densities in the natural numbers of two sets of numbers. These sets are
for integers .
Theorem 1. Let with and . Then the asymptotic density of the set is . In addition the asymptotic density of the set is .
We firstly look at . We have, for fixed ,
where is an error term introduced by not rounding down to the nearest integer. So, letting
where the new error term satisfies .
The proof for is very similar. With and defined as above we have for fixed ,