A question about numbers from Euclid's proof of infinitude of primes

The product of the first $75$ primes, plus $1$, is prime. (That number is $171962010545840643348334056831754301958457563589574256043877$ $110505832165523856261308397965147955578800999455782202456522$ $6932906295208262756822275663694111$.)

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(I misunderstood the question at this point. The poster wants to know if any prime appears more than once in any given entry of the sequence, not in any pair of entries in the sequence.)

$277$ is a factor of the seventh number ($510511$) and the seventeenth ($1922760350154212639071$).

There are more Euclid primes, but it isn't known if there are infinitely many. It's just conjectured, as well as all of Euclid numbers being squarefree: https://oeis.org/A006862