On a sequence involving sums of primes

For $n=1,2,3,\dots$ let $S_n$ be the sum of the first $n$ primes. We mainly show that the sequence $a_n=\text{root}n\text{of}{S}_n\mathrm{/}n(n=1,2,3,\dots )$ is strictly decreasing, and moreover the sequence $a_{n+1}\mathrm{/}{a}_n(n=10,11,\dots )$ is strictly increasing. We also formulate similar conjectures involving twin primes or partitions of integers.