Let a1, a2, . . . , an be a sequence of integers with values between 2 and 1995 such that:
(i) Any two of the ai’s are relatively prime,
(ii) Each ai is either a prime or a product of primes.
Determine the smallest possible values of n to make sure that the sequence will contain a prime number.
[APMO 1995 Q2][with minor edit of typo]
I am posting this question as I think it is interesting, however, I also feel it needs some interpretation. I have copied it as written (apart from one typo correction), but I'm not sure why it asks for "values of n" in the plural. I assume the question is asking us to firstly find the maximum number of terms such that conditions (i) and (ii) are satisfied but without any primes appearing; then by adding one unused prime to such a sequence we would have found the minimum number that guarantees that a prime be present. I also assume that the integers are 2 to 1995 inclusive. I leave these assumptions open for discussion; it may be a case of lost in translation as APMO is the Asian Pacific MO.
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