This was a collaborative effort from Vincent Diepeveen, Tony Reix, Paul Underwood and Jeff Gilchrist.
2^13380298-27 sieved up to 2e32 (Paul Underwood) and from 2e32 to 15e11 (Jeff Gilchrist, k1b2sieve 1.1 on Linux with Intel Xeon E5-2667 v4)
2^13380298-27 is base 3-Fermat PRP! (Jeff Gilchrist, LLR 3.8.23 on Linux with Intel Xeon E5-2667 v4)
2^13380298-27 is Fermat, Lucas and Frobenius PRP! (P = 5, Q = 5, D = 5) (Jeff Gilchrist, LLR 3.8.23 on Linux with Intel Xeon E5-2667 v4)
2^13380298-27 is base 3-Fermat PRP! (Jeff Gilchrist, LLR 3.8.23 on Windows 10 with Intel Core i7-6700K)
2^13380298-27 is Fermat, Lucas and Frobenius PRP! (P = 5, Q = 5, D = 5) (Jeff Gilchrist, LLR 3.8.23 on Windows 10 with Intel Core i7-6700K)
2^13380298-27 is 3-PRP! (Jeff Gilchrist, OpenPFGW for Windows 4.0.1 with Intel Core i7-6700K)
2^13380298-27 is 5-PRP! (Jeff Gilchrist, OpenPFGW for Windows 4.0.1 with Intel Core i7-6700K)
2^13380298-27 is 7-PRP! (Jeff Gilchrist, OpenPFGW for Windows 4.0.1 with Intel Core i7-6700K)
2^13380298-27 is 11-PRP! (Jeff Gilchrist, OpenPFGW for Windows 4.0.1 with Intel Core i7-6700K)
2^13380298-27 is 13-PRP! (Jeff Gilchrist, OpenPFGW for Windows 4.0.1 with Intel Core i7-6700K)
2^13380298-27 is 17-PRP! (Jeff Gilchrist, OpenPFGW for Windows 4.0.1 with Intel Core i7-6700K)
2^13380298-27 is 19-PRP! (Jeff Gilchrist, OpenPFGW for Windows 4.0.1 with Intel Core i7-6700K)
2^13380298-27 is 23-PRP! (Jeff Gilchrist, OpenPFGW for Windows 4.0.1 with Intel Core i7-6700K)
2^13380298-27 is PRP! [N-1 base 2 & 5, Brillhart-Lehmer-Selfridge] (Jeff Gilchrist, OpenPFGW for Windows 4.0.1 with Intel Core i7-6700K)
2^13380298-27 is a Fermat Probable prime! (llrCUDA, Jeff GilChrist, nVidia P100)
GWNUM, Linux, Paul Underwood, n=2^13380298-27 passes:-
x^(n + 1) == 1 (mod n, x^2 - 3*x + 1) [*]
(x + 1)^(n + 1) == 2 + 3 (mod n, x^2 - 3*x + 1) [*]
(x + 2)^(n + 1) == 5 + 2*3 (mod n, x^2 - 3*x + 1) [*]
(2*x)^((n + 1)/2) == 2 (mod n, x^2 - 3*x + 1) [*][**]
2^((n - 1)/2) == -1 (mod n) and (x + 1)^n == -x + 1 (mod n, x^2 - 2)
(x + 1)^(n + 1) == 3 (mod n, x^2 + 2) [***]
[*] JacobiSymbol(3^2 - 4, n) = -1
[**] 2*JacobiSymbol(2*(3 + 2), n) = 2
[***] n == 5 mod 8
Thanks to Jean Penné, George Woltman, and Mark Rodenkirch for the software used in this project.

To be completed...