finding sets of integers whose subsets do not sum to a prime: how to address the problem? This is sort of a variation of the Hasler sequence, which requires that the sum of a certain number of directly previous integers (in the sequence) isn't prime.
I started to wonder: Suppose the rule is that, for a set S containing the sequence of non prime counting numbes, [1,4,...N] , is there a limit to N such that the sum of any subset of S containing  K elements of S is not prime?
I'm going to write some code to test this for small-ish N and K, but could use some help learning how this would be proved/disproved analytically.
 A: So for $K=2$ we get $N=2$ because the sum of the first two non primes $1, 4$ is $5$ which is already prime. Similarly for $K=3$ we get $N=3$ because $1+4+6=11$ which is also prime. One can compute a few more terms and you will get that $N$ is at most 1 or 2 bigger than $K$.
For larger values of $K$ one should still expect an $N$ that is only marginally bigger than $K$ by a heuristic argument. Given $K$ if $N=K+m$ for some value of $m$ that there are $m \choose N$ ways to pick a subset of size $K$ (which is the same as not picking $m$ numbers) and you want every single one of these to not be a prime. This grows very quickly, for $K$ large and $m$ small this is on the order of $N^m$.
Getting a good explicit bound for $N$ in terms of $K$ is probably hard. Some version of the pigeon hole principle might give you a rough bound like $N < 2K$ or $N < K + \sqrt{K}$. Using the prime number theorem in the sense that the probability of a number of size $n$ being prime is $1/{\ln(n)}$ might give a heuristic argument for the correct asymptotic size of $N$ in terms of $K$ but making this into a rigorous upper bound is usually very hard.
