Consecutive Prime Gap Sum (Amateur) List of the first fifty prime gaps:
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4.
My conjecture is that the sum of consecutive prime gaps is always prime whenever a prime gap of 2 is added.
$$ 1 + 2 = 3 $$
$$ 1 + 2 + 2 = 5 $$
$$ 1 + 2 + 2 + 4 + 2 = 11 $$
$$ 1 + 2 + 2 + 4 + 2 + 4 + 2 = 17 $$
$$ 1 + 2 + 2 + 4 + 2 + 4 + 2 + 4 + 6 + 2 = 29 $$
I don't know if this is meaningful or how to go about testing it completely (I've tested it up to 461) so I'll just leave this here and see what comes of it.

Set $g_n=p_{n+1}-p_n$, where $p_n$ is the series of prime numbers, with $p_1=2$.
Then 
$$
p_1+\sum_{i=1}^n g_i=\sum_{i=1}^n g_i+2=p_{n+1}.
$$
So the conjecture is obviously true, but not useful.
