I am a hobby computer scientist and im looking for simple fast ways to generate gigantic primes, specifically the ones im describing in this post. So leave out prime factorization for a moment. Observe the primes: 2, 5, 17, 41, 197, 281, 7699, 8893,.. and so forth (A013918).
These are primes that are sums of other primes. OEIS has a list.
Example: Number 11 is not in the list because 2+3+5=10, 2+3+5+7=17. No other consecutive addition of primes add up to that prime. We can do the same for the other primes and find that they are not in the list.
I find these primes a special type in the sense described above. So my first question is this: Does these primes have an own name? Or are they just called prime sums? There's a small section about these specific primes here.
More info
The number of terms we need to get to that prime number (A013916): 1, 2, 4, 6, 12, 14, 60.. (further down i call these numbers weights)
$n: p_1+p_2..+p_n\\ 1: 2\\ 2: 2+3 = 5.\\ 4: 2+3+5+7 = 17.\\ 6: 2+3+..+13 = 41.\\ 12: 2+3+..+37 = 197.$
etc. where n is number of terms (the weight).
same as
$\sum_{i=1}^n p_n$
Info we should know
how many terms to add
all the consecutive primes up to that point
We know the number of additions are exactly even for each new prime.
Problem
Finding mersenne primes are computational very challenging (ref. mathworld). What about these specific primes in the list above? There should be a method to compute these primes by adding previous primes.
We do not need to find all the primes between one prime and the next: example 5,17: to get 17 we added 2+3+5 and 7. not 11 and 13.
What we need is a (preferably simple) way to generate a list of weights for these? (number of terms to add). If we know this we can compute gigantic primes, because we do not need to check if the number is a prime or not, right?
The numbers we are dealing with are non-negative integers, and of course primes. There is a way to add gigantic numbers that requires approx. one hundred megabyte of memory (which is approx. the space needed for generating primes up to to the largest mersenne prime found today) and should be really fast using a homogeneous structure.
Is there a sieve method to compute these primes (2,5,17,41..)? Or is there a way to compute the weights (number of terms) to get to those primes?
Primes are mysterious. No one knows when the next prime is gonna come. So I think when you can not predict the next prime, then how can you predict when the sum of primes is gonna be prime. I think you can not make the way unless we know the method to predict nth prime.