Algorithm for generating coprime number sequences?
Yes.  It's possible to satisfy all of your requirements, by combining some standard tools:


Let $F$ be a function that, given a sequence of random bits as input, outputs a random 1000-bit prime.  $F$ can be implemented by repeatedly picking a random 1000-bit number, then using a primality test to check if it is prime, and outputting the first prime number you see.
Let $G$ be a pseudorandom generator that expands a short random seed to a long sequence of pseudorandom numbers, e.g., AES-CTR mode.
Let $H$ be a cryptographic hash function, e.g., SHA256.
Let $s$ be a random seed.


Now define

$$f(i) = F(G(H(s,i))).$$

Then the sequence

$$f(1),f(2),f(3),f(4),f(5),\dots$$

satisfies all of your requirements:


Every number in the sequence is prime, and it is super-unlikely that any prime will repeat (this can be quantified using the birthday paradox), so all of the numbers in the sequence are co-prime.
It is super-unlikely that any number will be less than 50.  If you prefer an absolute guarantee, you can tweak the definition of $F$ to require that the number is both prime and at least your lower bound.
The sequence can be efficiently computed: each of the building blocks $F,G,H$ has efficient instantiations, so $f$ can be computed efficiently.  Moreover, this construction permits efficient random access into the sequence.