The corresponding indices k are 1, 2, 34, 107, 1568, 1933, 3551, 6793, ... (see A200821).
The generalization of this sequence is possible with the primes of the form (b^n + k)*b^n + 1.
For k = 107, a(4) has 65 digits;
for k = 1568, a(5) has 945 digits;
for k = 1933, a(6) has 1164 digits;
for k = 3551, a(7) has 2138 digits;
for k = 6793, a(8) has 4090 digits.
