
COMMENTS

The generalization of this sequence is possible with the primes of the form (b^n +k)*b^n +1.
The corresponding n are : 1, 3, 6, 14, 21, 27, 51, 61, 103, 123, 126, 414, 499, 1509, 2389,…
(see A200823).
For n = 103, a(9) has 63 digits;
for n = 123, a(10) has 75 digits;
for n = 126, a(11) has 76 digits;
for n = 414, a(12) has 250 digits;
for n = 499, a(13) has 301 digits;
for n = 1509, a(14) has 909 digits;
for n = 2389, a(15) has 1439 digits.


MATHEMATICA

a={}; Do[p=(2^n + n)*2^n+1; If[PrimeQ[p], AppendTo[a, p]], {n, 10^3}]; Print[a];
