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COMMENTS
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The generalization of this sequence is possible with the primes of the form (b^n +- k)*b^n +- 1.
The corresponding indices k are 1, 3, 6, 14, 21, 27, 51, 61, 103, 123, 126, 414, 499, 1509, 2389, ...
(see A200823).
For k = 103, a(9) has 63 digits;
for k = 123, a(10) has 75 digits;
for k = 126, a(11) has 76 digits;
for k = 414, a(12) has 250 digits;
for k = 499, a(13) has 301 digits;
for k = 1509, a(14) has 909 digits;
for k = 2389, a(15) has 1439 digits.
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MATHEMATICA
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a={}; Do[p=(2^n + n)*2^n+1; If[PrimeQ[p], AppendTo[a, p]], {n, 10^3}]; Print[a];
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