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

limunom623

limunom623

Answered question

2022-11-18

How many numbers less than x have a prime factor that is not 2 or 3
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 log 2 x
Or to make the same determination about 3:
x log 3 x
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 log 2 x log 3 x
and greater than:
x log 2 x log 3 x x 6
Thanks

Answer & Explanation

Faith Wise

Faith Wise

Beginner2022-11-19Added 17 answers

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
log ( 2 n ) log ( 3 n ) 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 ( n log n )

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?