Let $S_n=\Sigma^n_{k=1}p_k$, where $p_k$ is the $k$-th prime number.
Conjecture:
$$\forall p\in\mathbb P\exists n\in\mathbb N: p|S_n$$
Verified for the $1000$ first primes. Is there a proof for this result in general?
In the diagram primes are projected on the x-axis and $n$ (as in $S_n$) on the y-axis.
As H.H.Rugh commented there is a stronger conjecture for all positive integers m. Below a table of $n$-records for different $m\in\mathbb Z^+$ (some of them primes):
m n factorization of Sn
1 (1) 1 (2)
3 (3) 10 (3,43)
6 (2,3) 57 (2,3,5,229)
12 (2,2,3) 97 (2,2,3,1879)
18 (2,3,3) 113 (2,3,3,41,43)
35 (5,7) 180 (5,7,2531)
42 (2,3,7) 305 (2,2,2,3,7,7,239)
90 (2,3,3,5) 357 (2,2,2,3,3,3,5,367)
101 (101) 422 (5,101,1129)
137 (137) 861 (2,137,9739)
163 (163) 902 (5,7,11,47,163)
195 (3,5,13) 907 (2,3,3,5,13,2551)
202 (2,101) 1207 (2,2,19,101,719)
222 (2,3,37) 1359 (2,2,2,3,3,37,2671)
252 (2,2,3,3,7) 1683 (2,2,3,3,7,17,37,71)
326 (2,163) 1765 (2,2,3,23,163,277)
474 (2,3,79) 2077 (2,2,2,3,5,79,1861)
504 (2,2,2,3,3,7) 2133 (2,2,2,3,3,3,7,53,233)
522 (2,3,3,29) 2379 (2,3,3,3,3,3,7,29,239)
643 (643) 2529 (2,3,3,11,211,643)
647 (647) 3092 (11,647,5791)
658 (2,7,47) 3353 (2,3,7,43,47,577)
700 (2,2,5,5,7) 3593 (2,2,5,5,7,103,787)
817 (19,43) 4683 (2,3,11,19,43,1847)
995 (5,199) 5329 (2,3,5,17,199,1291)
1004 (2,2,251) 6415 (2,2,2,251,96643)
1204 (2,2,7,43) 6533 (2,2,2,2,7,7,31,43,193)
1459 (1459) 7241 (2,2,3,3,5,5,191,1459)
1488 (2,2,2,2,3,31) 7307 (2,2,2,2,2,3,31,85909)
1610 (2,5,7,23) 8079 (2,5,7,23,43,4567)
1677 (3,13,43) 10171 (2,2,2,3,3,3,13,43,4259)
1870 (2,5,11,17) 10331 (2,5,11,17,71,4003)
2035 (5,11,37) 11459 (2,2,3,3,5,11,37,9029)
2616 (2,2,2,3,109) 11753 (2,2,2,3,3,3,3,3,37,89,109)
2672 (2,2,2,2,167) 18137 (2,2,2,2,7,167,93047)
3420 (2,2,3,3,5,19) 21709 (2,2,3,3,3,5,13,19,137,139)
3830 (2,5,383) 27617 (2,5,53,383,20749)
4232 (2,2,2,23,23) 38861 (2,2,2,2,3,3,23,23,113189)
7394 (2,3697) 45381 (2,107,3697,15083)
7450 (2,5,5,149) 47323 (2,3,3,5,5,7,41,149,677)