Consider sequence A110293 in Sloane. The first 4 entries, a(1) to a(4), 7, 13, 97, 181, are all prime. However, a(5) =1351 is compound; it is 7*193. Although 7 is not a new factor in the series, 193 is, because all the earlier entries (other than 1) were prime.
A prime entry, e.g. 7, or a prime factor of a compound entry, that is new to the sequence, e.g. 193, is called a primitive prime divisor, or PPD.
Claim: every entry in the sequence contains at least one PPD.
Prove it, or find a counter-example.