
COMMENTS

All terms in the sequence fit the pattern 11m + 1 or 33m  1 up to a(21) = 177. Note that 10*(11m + 1)^k + 1 is divisible by 11 for all k, and 10*(33m  1)^k + 1 is divisible by 3 when k is odd and 11 when k is even.
A prime 10*173^k+1 is now known (for k=264234, see link). The next blocker is 10*185^k+1 (is 10 a base185 Sierpiński number?).  Jeppe Stig Nielsen, Apr 30 2018
10*185^k+1 is composite for all k <= 10^6 (see the Barnes link).  Eric Chen, Jun 07 2018
If a prime 10*185^k+1 were found, then this sequence would continue with 188, 197, 199, 210, 221, 230, 232, 243, 254, 263, 265, 276, 287, 296, 298, 309, 320, 329, 331, ...  Eric Chen, Jun 07 2018
The prime 10*173^264234+1 is found to solve the generalized Sierpinski problem in base 173 (with conjectured smallest Sierpinski number k=28, see the Barnes link and A123159 for these problems).  Eric Chen, Jun 07 2018
All unknown terms below 1024 are 185, 338, 417, 432, 537, 614, 668, 743, 744, 773, 786, 827, 863, 929, 935, 977, 986, 1000, 1004. Search limits: 185 at 10^6, 417 at 4*10^5, 743, 773 and 935 at 2*10^5, 338, 744 and 977 at 10^5, 432 at 25000, other numbers except 1000 at 5000, 1000 is corresponding the generalized Fermat prime in base 10 and already searched to (2^241)/3, since the smallest prime of the form 10^n+1 greater than 101 is at least 10^(2^24)+1.  Eric Chen, Jun 09 2018
Large primes with n <= 1024 and exponent > 10^4: 10*173^264234+1, 10*198^47664+1, 10*311^314806+1, 10*341^106008+1, 10*449^18506+1, 10*492^42842+1, 10*605^12394+1, 10*708^17562+1, 10*710^31038+1, 10*800^15104+1, 10*802^149319+1, 10*879^25003+1, they are all proven primes, i.e., not merely probable primes, since they can be proved prime with the N1 method.  Eric Chen, Jun 09 2018
All other n <= 1024 with n != 1 (mod 11) and n != 32 (mod 33) have at least one prime of the form 10*n^k+1 with k <= 10^4.  Eric Chen, Jun 09 2018
