The longest sum of consecutive primes that add to a prime less than 1,000,000 
In Project Euler problem $50,$ the goal is to find the longest sum of consecutive primes that add to a prime less than $1,000,000. $

I have an efficient algorithm to generate a set of primes between $0$ and $N.$
My first algorithm to try this was to brute force it, but that was impossible.
I tried creating a sliding window, but it took much too long to even begin to cover the problem space.
I got to some primes that were summed by $100$+ consecutive primes, but had only run about $5$-$10$% of the problem space.
I'm self-taught, with very little post-secondary education.

Where can I read about or find about an algorithm for efficiently calculating these consecutive primes?
I'm not looking for an answer, but indeed, more for pointers as to what I should be looking for and learning about in order to solve this myself.


I used the mathematica code:
list = {};
Do[
 a = Sum[Prime[i], {i, k, j}];
 If[900000 < a <= 1000000 && PrimeQ[a],
  AppendTo[list, {k, j, a}]],
 {k, 1, 1000}, {j, 1, 1000}]

and then.
Sort[list, #1[[3]] < #2[[3]] &].

This seems very inefficient, but it was pretty quick. I found that among the sequences of the first 1000 primes, the greatest prime sum was from 459 to 695, with a value of 999749. This is probably not what you're looking for since I don't really consider mathematica to be programming.
Btw, would it be an interesting to ask how many ways a number can be summed as the sum of consecutive primes?
