Task

Considering in order of length, n, all sequences of consecutive
primes, p, from 2 onwards, where p < 1000 and n>0, select those
sequences whose sum is prime, and for these display the length of the
sequence, the last item in the sequence, and the sum.




p = Prime[Range[PrimePi[1000]]];
TableForm[
 Select[Transpose[{Range[Length[p]], p, Accumulate[p]}], Last /* PrimeQ], 
 TableHeadings -> {None, {"Prime count", "Prime", "Prime sum"}}
]