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The sum of the first n primes often divides their product, but it often doesn't. Which of the two happens most often? Any references to this question?

Bernardo Recamán Santos
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  • It's got become less likely as $n$ increases, because more primes between $p_n$ and $\sum_n {p_i}$ become available to be factors of the sum that won't divide the product, and $(\sum_n {p_i})/p_n$ grows without limit. Plus there is no guarantee that the sum will be square-free, unlike the product (e.g. the sum may be divisible by $4$). – Joffan Jan 26 '21 at 15:38
  • I think, we can answer this only by counting the cases upto some large limit. – Peter Jan 26 '21 at 15:46
  • I am now told this is A051838 in the oeis. – Bernardo Recamán Santos Jan 26 '21 at 16:06
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    It seems like from the table that about 2 out of 9 do and 7 out of 9 do not. That's examining the first 45 thousand primes. The question can still be asked, does the limit exist and is it 0 or something else? – John L Jan 26 '21 at 16:16
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    For $n=10^7$ (the first $10$ million primes) the ratio of cases where the divisibility holds is $$0.2174082$$ which is smaller than $$0.218721$$ , the ratio for $n=10^6$. This could indicate a slow descent towards $0$. With the theory of smooth numbers, we might be able to get reasonable estimates. – Peter Jan 29 '21 at 11:49

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