Does there exist a strictly increasing sequence of primes $(q_i)_{i \in \mathbb N}$ such that $\text {ds} (\prod_{k=1}^l q_k)$ is prime for every $l \in \mathbb N$?

Here $\text{ds}(n)$ denotes a digit sum of $n$.

I do not know from where should I attack this problem, main thoughts are:

a) If we start with some prime then there is probably an infinite number of primes that will not work with that starting-point prime, so that still leaves us with probably an infinite number of primes that we could choose as to be our second prime. Then those two first primes again probably determine an infinite number of primes that will not work but again probably leave us with an infinite number of primes that could serve as a third prime to choose. So, maybe some sieving could do the job, if there are some specialized sieves for this type of problems.

b) On the other hand, every prime that does not have digit-sum a composite number seems as a suitable candidate to be a starting point because there are so many possible multiplications that we can do here that it would be a sort of a miracle if some primes are better-behaved than some others as a candidates for a starting point.

c) If we choose some starting-point prime a reason why an infinite number of primes will not be suitable as a second prime is because probably it is often that digit sum of product of two primes is a composite number, and this should also hold for products of $l$ primes, for every $l \in \mathbb N \setminus \{1\}$.

d) All of this leads to a conclusion that if every prime that does not have digit-sum a composite number could be a suitable starting-point prime that it could also be that sequence associated to that prime could be unique and strictly determined by a starting point, but also it could be that it is not so.

Also, in an answer or in a comment, I would welcome if someone gives me some works (books/articles) on "digit-sum theory" because it seems to me even if there are glimpses of such a theory that such a theory is in a very poor stage of a development

`genseq[n_, m_] := genseq[n, m] = If[n == 1, If[Not[PrimeQ[Total[IntegerDigits[Prime[m]]]]], {}, {Prime[m]}], Module[{s = genseq[n - 1, m], pr, k}, pr = Times @@ s; If[Not[PrimeQ[Total[IntegerDigits[pr]]]], Return[{}]]; k = PrimePi[Last[s]] + 1; While[Not[PrimeQ[Total[IntegerDigits[pr Prime[k]]]]], k++]; Append[s, Prime[k]] ] ]`

$\endgroup$ – მამუკა ჯიბლაძე Mar 23 '18 at 19:44