$\text{Statement}$
Denote $p_n$ the $n$th prime. Consider the set $$\mathcal{P}=\{\sum_{i=1}^np_i:n\in\mathbb{N}\}$$ Prove that $|\mathcal{P}\cap\mathbb{P}|=\infty$
$\text{Why am I studying this}$
First of all, I find it an itriguing and rather unusual result. I also want to see how we can generate primes by adding numbers, so this was a natural idea. To conclude, I also highly highly suspect these primes have some extra properties from normal ones.
$\text{Computational arguments}$
First of all, here are the first $200$ values for $\sum_{i=1}^{n}p_i$:
5 10 17 28 41 58 77 100 129 160 197 238 281 328 381 440 501 568 639 712 791 874 963 1060 1161
1264 1371 1480 1593 1720 1851 1988 2127 2276 2427 2584 2747 2914 3087 3266 3447 3638 3831 4028
4227 4438 4661 4888 5117 5350 5589 5830 6081 6338 6601 6870 7141 7418 7699 7982 8275 8582 8893
9206 9523 9854 10191 10538 10887 11240 11599 11966 12339 12718 13101 13490 13887 14288 14697
15116 15537 15968 16401 16840 17283 17732 18189 18650 19113 19580 20059 20546 21037 21536 22039
22548 23069 23592 24133 24680 25237 25800 26369 26940 27517 28104 28697 29296 29897 30504 31117
31734 32353 32984 33625 34268 34915 35568 36227 36888 37561 38238 38921 39612 40313 41022 41741
42468 43201 43940 44683 45434 46191 46952 47721 48494 49281 50078 50887 51698 52519 53342 54169
54998 55837 56690 57547 58406 59269 60146 61027 61910 62797 63704 64615 65534 66463 67400 68341
69288 70241 71208 72179 73156 74139 75130 76127 77136 78149 79168 80189 81220 82253 83292 84341
85392 86453 87516 88585 89672 90763 91856 92953 94056 95165 96282 97405 98534 99685 100838
102001 103172 104353 105540 106733 107934 109147 110364
These are the primes that were generated:
2, 5, 17, 41, 197, 281, 7699, 8893, 22039, 24133, 25237, 28697, 32353, 37561, 38921, 43201, 44683, 55837, 61027, 66463, 70241, 86453, 102001, (and some others: 109147, 116533, 119069, 121631, 129419, 132059, 263171, 287137, 325019, 329401, 333821, 338279, 342761)
$\text{Generalizations}$
Of course, there are many ways to generalize this. I will state 3, which might be of interest. However, for the sake of simplicity, i will only leave the first statement as a conjecture. These new results are quite similar so I think a hypothetical proof would not differ too much.
$\text{Generalization } 1:$
The same question as the original one, but this time $$\mathcal{P_k}=\{\sum_{i=k}^np_i:n\in\mathbb{N}\}$$ Prove that $\forall k$, $|\mathcal{P_k}\cap\mathbb{P}|=\infty$
$\text{Generalization } 2:$
The same question as the original one, but this time $$\mathcal{P^l}=\{\sum_{i=1}^np_i^l:n\in\mathbb{N}\}$$ Prove that $\forall k$, $|\mathcal{P^l}\cap\mathbb{P}|=\infty$
$\text{Generalization } 3:$
A combination of generalization $1$ and $2$ $$\mathcal{P_k^l}=\{\sum_{i=k}^np_i^l:n\in\mathbb{N}\}$$ Prove that $\forall k$, $|\mathcal{P_k^l}\cap\mathbb{P}|=\infty$
I am not providing computational arguments for these ones.