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Question Q1 seems to me an extremely hard problem, but I also believe that the answer is affirmative because the heuristic argument exposed by David Speyer.

I studied with FLorian Luca [1] a related problem that could help to answer question Q2:

Let $A$ the set of integers $n$ such that the sum of the first n primes is divisible by $n$ (instead of $p_n$). In other words, $A$ is the set of $n$ such that the arithmetic mean of the first $n$ primes is an integer:

$A=\{ n: \frac{p_1+\cdots +p_n}n \in \mathbf Z \}=\{ 1, 23, 53, 853, 11869, 117267, 339615, 3600489,..\} $

These numbers are not so rare because in this case the heuristic gives $ A(x)\asymp \sum_{n\le x}\frac 1n\sim \log x$.

We could not prove that $A$ has infinite elements but we proved that they are rare:

$$A(x)\ll x e^{-C(\log x)^{3/5}(\log \log x)^{-1/5}}.$$ Later Matomaki [2] proved the stronger estimate, $A(x)\ll x^{\frac{19}{24}+\epsilon}$.

[1] Cilleruelo, Javier; Luca, Florian, On the sum of the first n primes. Q. J. Math. 59 (2008), no. 4. http://www.uam.es/personal_pdi/ciencias/cillerue/

[2] Matomäki, Kaisa, A note on the sum of the first n primes. Q. J. Math. 61 (2010), no. 1,

Question Q1 seems to me an extremely hard problem, but I also believe that the answer is affirmative because the heuristic argument exposed by David Speyer.

I studied with FLorian Luca [1] a related problem that could help to answer question Q2:

Let $A$ the set of integers $n$ such that the sum of the first n primes is divisible by $n$ (instead of $p_n$). In other words, $A$ is the set of $n$ such that the arithmetic mean of the first $n$ primes is an integer:

$A=\{ n: \frac{p_1+\cdots +p_n}n \in \mathbf Z \}=\{ 1, 23, 53, 853, 11869, 117267, 339615, 3600489,..\} $

These numbers are not so rare because in this case the heuristic gives $ A(x)\asymp \sum_{n\le x}\frac 1n\sim \log x$.

We could not prove that $A$ has infinite elements but we proved that they are rare:

$$A(x)\ll x e^{-C(\log x)^{3/5}(\log \log x)^{-1/5}}.$$ Later Matomaki [2] proved the stronger estimate, $A(x)\ll x^{\frac{19}{24}+\epsilon}$.

[1] Cilleruelo, Javier; Luca, Florian, On the sum of the first n primes. Q. J. Math. 59 (2008), no. 4. http://www.uam.es/personal_pdi/ciencias/cillerue/articulos.html

[2] Matomäki, Kaisa, A note on the sum of the first n primes. Q. J. Math. 61 (2010), no. 1,

Question Q1 seems to me an extremely hard problem, but I also believe that the answer is affirmative because the heuristic argument exposed by David Speyer.

I studied with FLorian Luca [1] a related problem that could help to answer question Q2:

Let $A$ the set of integers $n$ such that the sum of the first n primes is divisible by $n$ (instead of $p_n$). In other words, $A$ is the set of $n$ such that the arithmetic mean of the first $n$ primes is an integer:

$A=\{ n: \frac{p_1+\cdots +p_n}n \in \mathbf Z \}=\{ 1, 23, 53, 853, 11869, 117267, 339615, 3600489,..\} $

These numbers are not so rare because in this case the heuristic gives $ A(x)\asymp \sum_{n\le x}\frac 1n\sim \log x$.

We could not prove that $A$ has infinite elements but we proved that they are rare:

$$A(x)\ll x e^{-C(\log x)^{3/5}(\log \log x)^{1/5}}.$$ Later Matomaki [2] proved the stronger estimate, $A(x)\ll x^{\frac{19}{24}+\epsilon}$.

[1] Cilleruelo, Javier; Luca, Florian, On the sum of the first n primes. Q. J. Math. 59 (2008), no. 4. http://www.uam.es/personal_pdi/ciencias/cillerue/

[2] Matomäki, Kaisa, A note on the sum of the first n primes. Q. J. Math. 61 (2010), no. 1,

Question Q1 seems to me an extremely hard problem, but I also believe that the answer is affirmative because the heuristic argument exposed by David Speyer.

I studied with FLorian Luca [1] a related problem that could help to answer question Q2:

Let $A$ the set of integers $n$ such that the sum of the first n primes is divisible by $n$ (instead of $p_n$). In other words, $A$ is the set of $n$ such that the arithmetic mean of the first $n$ primes is an integer:

$A=\{ n: \frac{p_1+\cdots +p_n}n \in \mathbf Z \}=\{ 1, 23, 53, 853, 11869, 117267, 339615, 3600489,..\} $

These numbers are not so rare because in this case the heuristic gives $ A(x)\asymp \sum_{n\le x}\frac 1n\sim \log x$.

We could not prove that $A$ has infinite elements but we proved that they are rare:

$$A(x)\ll x e^{-C(\log x)^{3/5}(\log \log x)^{-1/5}}.$$ Later Matomaki [2] proved the stronger estimate, $A(x)\ll x^{\frac{19}{24}+\epsilon}$.

[1] Cilleruelo, Javier; Luca, Florian, On the sum of the first n primes. Q. J. Math. 59 (2008), no. 4. http://www.uam.es/personal_pdi/ciencias/cillerue/

[2] Matomäki, Kaisa, A note on the sum of the first n primes. Q. J. Math. 61 (2010), no. 1,

Question Q1 seems to me an extremely hard problem, but I also believe that the answer is affirmative because the heuristic argument exposed by David Speyer.

I studied with FLorian Luca [1] a related problem that could help to answer question Q2:

Let $A$ the set of integers $n$ such that the sum of the first n primes is divisible by $n$ (instead of $p_n$). In other words, $A$ is the set of $n$ such that the arithmetic mean of the first $n$ primes is an integer:

$A=\{ n: \frac{p_1+\cdots +p_n}n \in \mathbf Z \}=\{ 1, 23, 53, 853, 11869, 117267, 339615, 3600489,..\} $

These numbers are not so rare because in this case the heuristic gives $ A(x)\asymp \sum_{n\le x}\frac 1n\sim \log x$.

We could not prove that $A$ has infinite elements but we proved that they are rare:

$$A(x)\ll x e^{-C(\log x)^{3/5}(\log \log x)^{1/5}}.$$ Later Matomaki [2] proved the stronger estimate, $A(x)\ll x^{\frac{19}{24}+\epsilon}$.

[1] Cilleruelo, Javier; Luca, Florian, On the sum of the first n primes. Q. J. Math. 59 (2008), no. 4.

[2] Matomäki, Kaisa, A note on the sum of the first n primes. Q. J. Math. 61 (2010), no. 1,

Question Q1 seems to me an extremely hard problem, but I also believe that the answer is affirmative because the heuristic argument exposed by David Speyer.

I studied with FLorian Luca [1] a related problem that could help to answer question Q2:

Let $A$ the set of integers $n$ such that the sum of the first n primes is divisible by $n$ (instead of $p_n$). In other words, $A$ is the set of $n$ such that the arithmetic mean of the first $n$ primes is an integer:

$A=\{ n: \frac{p_1+\cdots +p_n}n \in \mathbf Z \}=\{ 1, 23, 53, 853, 11869, 117267, 339615, 3600489,..\} $

These numbers are not so rare because in this case the heuristic gives $ A(x)\asymp \sum_{n\le x}\frac 1n\sim \log x$.

We could not prove that $A$ has infinite elements but we proved that they are rare:

$$A(x)\ll x e^{-C(\log x)^{3/5}(\log \log x)^{1/5}}.$$ Later Matomaki [2] proved the stronger estimate, $A(x)\ll x^{\frac{19}{24}+\epsilon}$.

[1] Cilleruelo, Javier; Luca, Florian, On the sum of the first n primes. Q. J. Math. 59 (2008), no. 4. http://www.uam.es/personal_pdi/ciencias/cillerue/

[2] Matomäki, Kaisa, A note on the sum of the first n primes. Q. J. Math. 61 (2010), no. 1,