Primes in AP

Number Theory Level pending

Find all triplets of the form \(p_1,n,p_2\) where \(p_1,p_2\) are Twin primes and \(n\), their arithmetic mean is a Triangular number.

Enter your answer as sum of the solutions \(\sum (p_i+n_i+p_{2i})\) .

Explicit example : Say if you've got the solutions as \((1,2,3)\) and \((29,30,31)\) then your answer would be \(1+2+3+29+30+31=96\)

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