Partial sums of primes

$$2+3+5+7+11+13...$$ is clearly the sum of the primes.

Now I consider partial sums such:

$$2+3+5+7+11=28$$ which is divisible by $$7$$

My question is:

are there infinitely many partial sums such that:

$$p_1+p_2+p_3+...+p_{k}+p_{k+1}=m*p_{k}?$$ with $$m$$ some positive integer? With Pari/gp apparently up to 10^10 there are only two examples $$7$$=$$p_k$$ and $$8263=p_k$$. Heuristically do you think that infinitely many such partial sums should exist? Note: 7 and 8263 are both primes belonging to primes on the left side of the triangle formed by listing successively the prime numbers in a triangular grid. See https://oeis.org/A078721 Note in both cases $$2+3+5+7=17$$ is prime and $$2+3+5+...+p_{1036}=3974497$$ is prime. I note that $$17$$ and $$3974497$$ are primes of the form $$4s+1$$, whereas $$p_4=7$$ and $$p_{1036}=8263$$ are primes of the form $$6s+1$$. $$7$$ and $$8263$$ are primes such that starting from the right, the odd positioned digits are prime and the even positioned digits are composite. But also $$5$$ and $$8243$$ which are the previous primes have this property. No other prime of this type found below $$10^{12}$$ I noticed that 7! has 4 digits where 4 is a palindrome. 8263! has 28782 digits where 28782 is a palindrome.

• Strongly related: mathoverflow.net/questions/120511/…. Also crossposted on MSE: math.stackexchange.com/questions/3161810/23571113 (please don't do this anymore). – Alex M. Mar 25 at 22:31
• Seven edits in the last 12 hours. – Gerry Myerson Mar 26 at 21:11
• Now up to Version 13. – Gerry Myerson Mar 27 at 21:35
• I find these frequent edits go against the purpose of this forum. If you want to record frequent observations on a daily basis (whether they are significant or not), start a blog. You have asked a main question and gotten a reasonable answer; now move on. The numerology associated with the problem does not belong here. Next week, if you find a third prime satisfying the relations, you can report that here. Gerhard "Know When To Fold 'Em" Paseman, 2019.03.28. – Gerhard Paseman Mar 28 at 18:54
• Version 16. Please, homunc, give it a rest. – Gerry Myerson Mar 28 at 21:30

There is an heuristic argument that infinitely many such partial sums should exist. Consider $$P(k)$$, an heuristic estimate of the probability that the partial sum of the first $$k+1$$ primes would be divisible by $$p_k$$. Now $$p_k \sim k \log k$$ and if only random chance were involved, $$P(k) \approx \frac1{p_k} \sim \frac1{k \log k}$$
In that case, the expected number of primes with the property you want would be something like $$\int_2^\infty \frac1{x \log x}\,dx$$ and that integral diverges to infinity.
The reason it seems so rare is that the rate of divergence is like $$\log(\log x)$$ and while that function goes to infinity, "nobody ever sees it do so."
On the other hand, proving that there an infinite number of such values of $$k$$ (in the same sense that Euclid's argument proves there is no last prime) is probably quite difficult. And if the conjecture that there are an infinite number of such values of $$k$$ turned out to be false, proving that some particular $$k$$ is the last one with this property would seem to be even harder.