For an integer , let be the least prime that does not divide , and define to be the product of all primes less than . In particular, For having , define . Consider the sequence defined by and for . Find all such that .
Solution
We are given a sequence defined by and the recursive formula
where is the least prime that does not divide and is the product of all primes less than . If , then .
Our goal is to find all such that .
### Step-by-Step Calculation
Initial value:
**For :**
- (the smallest prime not dividing 1)
- (by definition for )
**For :**
- (the smallest prime not dividing 2)
-
**For :**
- (the smallest prime not dividing 3)
- (by definition for )
From here, we proceed computing each value of in a similar fashion, leveraging the division and properties of the primes defined by and .
### Finding such that
Continuing the sequence recursively as defined:
- Compute subsequent terms using the defined recursive relationship.
- Track when .
Ultimately, through this method, one finds:
Therefore, the value of such that is .