I thought (hoped) at the time that I had coined a type of prime number that was of the form 2p+q where both p and q are prime. I named them "Aidian" primes after my girlfriend.

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Investigate the distribution of these primes. Frequency within a range.

Frequency with p and q less than a certain number and p and q both varying freely out of the primes. Make sure 2p+q is also prime. Then add 2p+q to the list. So you're checking each prime number to see if it can be written in the form 2p+q for any other two primes.

Frequency with p and q less than a certain number and p and q be consecutive. Everything the same as above but then you're counting how many primes can be written in the form 2p+q where p and q are consecutive (original question).

Frequency with q varying freely under a certain number and let p be 1,2,3 and try 4,5. q+2 primes are called twin. q+4 are called cousin. q+6 are called sexy primes. Looking at the frequency for each of these.

Sum of 1/x where x is 2p+q p,q following each of the conditions above. Try to show that this sum approaches infinity. My guess is that we could compare the rate of the model to the log(n) function, and if the rate is greater, then this would be evidence that the sum 1/x diverges.

Things to consider:

Not every Aidian prime has a unique p,q such that 2p+q=the prime. For instance, 17=2*7+3 and also 17=2*2+13. If we exclude 2, does every Aidian prime have a unique p,q pair? How about tallying all of the p,q pairs that work for a prime.

What about any linear combination of two primes, p,q. What kind of results would this yield?