# Asymptotic densities of certain sets of numbers.

This note will establish formulae for the asymptotic densities in the natural numbers of two sets of numbers. These sets are

and

for integers .

**Theorem 1.** Let with and . Then the asymptotic density of the set is . In addition the asymptotic density of the set is .

**Proof.**

We firstly look at . We have, for fixed ,

where is an error term introduced by not rounding down to the nearest integer. So, letting

,

we have

where the new error term satisfies .

Then

Since and

.

The proof for is very similar. With and defined as above we have for fixed ,

and

and

.

**.**

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