Can we remove any prime number with this strange process?

Solution 1:

I have no proof so far but also written a program that checked that the only cyclic primes among the first $10^5$ steps are $p = 269$ in $S(58) = 6187$, and $p = 94 793$ in $S(9141) = 377 844 898$, where I call $S(n)$ the value of the sum at step $n$ (after adding the "next larger prime", not considering the removal of terms as a step, with $S(1) = 2$ which can be seen as result of adding $2$ to the empty sum $S(0)$).

FWIW, I get \begin{align}S(1000) &= 3\,362\,713\\S(2000) &= 14\,797\,503\\S(5000) &= 105\,157\,142\\S(10\,000) &= 456\,622\,646\\S(20\,000) &= 1\,979\,852\,987\\S(50\,000) &= 13\,618\,461\,808\\S(10^5) &= 58\,316\,786\,321.\end{align}

The smallest prime in the sum at these points is \begin{align}p(1000) &= 457\\p(2000) &= 509\\p(5000) &= 1451\\p(10\,000) &= 2281\\p(20\,000) &= 4129\\p(50\,000) &= 10\,631\\p(77\,000) &= p(10^5) = 16\,903.\end{align}

To be precise, my program considers a prime cyclic if it would give twice the same result, $\widehat S(n+1) = S(n)$, if it were used, which is then however avoided by using the next larger prime. I agree that this may not be the conceptually most satisfying definition... On the other hand, $p_{k+1}\mid\sum p_i$ isn't complete either (as mentioned in a comment), because depending on the smaller primes removed at that step or the next one, this does not necessarily give a loop. [However, if I'm not wrong, it's excluded that the next smaller prime is a factor of the sum at the same time as the largest prime $p_{k+1}$, because the sum is at most $\sim p_{k+1}^2/2 < p_{k}\cdot p_{k+1}$.]

The experimental results seem to confirm that eventually all primes will be removed. I agree with the heuristic argument that, if there's no particular modular restriction on the sum, then any prime factor will eventually occur (even infinitely often), and thus be removed from the sum. Given the "sequential" way of adding one prime after the other to the sum, there appears indeed no reason that from some point on, a particular prime factor should never occur again. This would yield the required result, but again, it's not a proof.

PS: I proposed the sequence $S(n)$ as oeis.org/A332198, where my program can be found.