How many numbers less than $x$ have a prime factor that is not $2$ or $3$

Question

I am trying to figure out the number of integers greater than $1$ and less than or equal to $x$ that have a prime factor other than $2$ or $3$. For example, there are only two such integer less than or equal to $7$.

It is straight forward to determine how many many integers less than or equal to $x$ have a prime factor other than $2$: $$x - \left\lfloor{\log}_2x\right\rfloor$$

Or to make the same determination about $3$: $$x - \left\lfloor{\log}_3x\right\rfloor$$

What is the method or formula for figuring out how many integers less than or equal to $x$ have a prime factor other than $2$ or $3$?

I know that it is less than:
$$x - \left\lfloor{\log}_2x\right\rfloor - \left\lfloor{\log}_3x\right\rfloor$$

and greater than: $$x - \left\lfloor{\log}_2x\right\rfloor - \left\lfloor{\log}_3x\right\rfloor - \left\lfloor\frac{x}{6}\right\rfloor$$

Thanks,

-Larry

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Edit: Added a greater than clause.

Answer 1

In Hardy's book of Twelve Lectures on Ramanujan's work, in the chapter "A lattice point problem", he discusses Ramanujan's result that

"the number of numbers of the form $2^x 3^y$ less than $n$ is

$\dfrac{\log(2n) \log(3n)}{2 \log 2 \log 3} $"

There is a very extended discussion of this problem. Among the results is a proof that the error in Ramanujan's formula is $O(\frac{n}{\log n})$

Answer 2

The answer is $n - A(n)$ where $A(n)$ is the number of integers $\le n$ of the form $2^x 3^y$. Now $A(n)$ is the number of nonnegative integer solutions of $x \log 2 + y \log 3 \le \log n$, i.e. the number of lattice points in the triangle $x \log 2 + y \log 3 \le \log n$, $x \ge 0$, $y \ge 0$. This is within $O(\log n)$ of the area of the triangle, i.e. $\log(n)^2/(2 \log(2)\log(3))$. But I doubt you'll get a "closed form" for the exact value.

EDIT: See also OEIS sequence A071521 and references there.