@ The Masked Avenger: In you powers of 3 example, $3^i$s are not coprime.

@ The Masked Avenger: Yes, you are right. Thank you very much.

@Gerry Myerson: Here is my motivation: I want to give an abstract formulation of Eratosthenes's Sieve and try to conject a property of Eratosthenes's Sieve. If this property are right, then I can use it to find a low bound of numbers of primes between any segment $[N, M+N]$ in the set of natural numbers, so this problem is connected to distribution of primes. You can view the following $d_i$s and $x$ in the conjecture as primes.

@Gerry Myerson, $d_i$ are coprime to each other, and they also coprime to $x$.

@ Gerry Myerson, yes, you are right, thanks. It is a mistake.