User blog comment:Pellucidar12/Idea for a function (Hyperprime counting functions)/@comment-30754445-20170723221034/@comment-1605058-20170723230550

For the record, this paper contains an infinitude proof.

Also, let me just mention that it is a big open problem whether there are infinitely many prime numbers with digit sum 2. We do know that out of all numbers with digit sum 2 (which are precisely the numbers 10k+1), almost all are composite. There is a big danger is fully trusting heuristic arguments like yours - sometimes the sets carry more structure than it seems at first, and this structure might force the numbers to be composite.