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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 $3$:

$$x – \left\lfloor{\log}_3x\right\rfloor$$

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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

Edit: Added a greater than clause.

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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

$O(\frac{n}{\log n})$

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.

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