Smallest dual Sierpinski/Riesel number in base 15$15$

For integer b≥2$b\ge 2$, We define:

• Sierpinski number base b$b$: Positive integer k$k$ such that gcd(k+1,b−1)=1$gcd(k+1,b-1)=1$ and k⋅bn+1$k\cdot b^n+1$ is composite for all integer n≥1$n\ge 1$
• Riesel number base b$b$: Positive integer k$k$ such that gcd(k−1,b−1)=1$gcd(k-1,b-1)=1$ and k⋅bn−1$k\cdot b^n-1$ is composite for all integer n≥1$n\ge 1$
• Dual Sierpinski number base b$b$: Positive integer k$k$ such that gcd(k,b)=1$gcd(k,b)=1$ and gcd(k+1,b−1)=1$gcd(k+1,b-1)=1$ and bn+k$b^n+k$ is composite for all integer n≥1$n\ge 1$
• Dual Riesel number base b$b$: Positive integer k$k$ such that gcd(k,b)=1$gcd(k,b)=1$ and gcd(k−1,b−1)=1$gcd(k-1,b-1)=1$ and bn−k$b^n-k$ is composite for all integer n≥1$n\ge 1$ such that bn>k+1$b^n>k+1$

See http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm for the (conjectured) smallest Sierpinski number base b$b$, and see http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm for the (conjectured) smallest Riesel number base b$b$.

(of course, most dual Sierpinski/Riesel conjectures are difficult to prove (like the original Sierpinski/Riesel conjectures), and unlike the original Sierpinski/Riesel conjectures (where the primes of the form k⋅bn+1$k\cdot b^n+1$ or k⋅bn−1$k\cdot b^n-1$ can be proven primes using the N−1$N-1$ or N+1$N+1$ test), the large primes of the dual Sierpinski/Riesel conjectures (i.e. the primes of the form bn+k$b^n+k$ or bn−k$b^n-k$) will only be PRP (probable prime) since neither N−1$N-1$ or N+1$N+1$ are smooth).

If a number k$k$ is Sierpinski number base b$b$ and gcd(k,b)=1$gcd(k,b)=1$, then k$k$ is also a dual Sierpinski number base b$b$, and if a number k$k$ is Riesel number base b$b$ and gcd(k,b)=1$gcd(k,b)=1$, then k$k$ is also a dual Riesel number base b$b$, since they have the same covering set, the exception is when the prime itself is in the covering set, e.g. 74$74$ is a Riesel number base 9$9$ but not dual Riesel number base 9$9$, since its covering set is {5,7,13,73}$\{5,7,13,73\}$, but 92−74$9^2-74$ is exactly 7$7$ and the (conjectured) smallest dual Riesel number base 9$9$ is 18404$18404$, also, 11$11$ is a Riesel number base 14$14$ but not dual Riesel number base 14$14$, since its covering set is {3,5}$\{3,5\}$, but 141−11${14}^1-11$ is exactly 3$3$ and the (conjectured) smallest dual Riesel number base 14$14$ is 19$19$, also, 13$13$ is a Riesel number base 20$20$ but not dual Riesel number base 20$20$, since its covering set is {3,7}$\{3,7\}$, but 201−11${20}^1-11$ is exactly 7$7$ and the (conjectured) smallest dual Riesel number base 20$20$ is 29$29$, also, 122$122$ is a Sierpinski number base 107$107$ but not dual Sierpinski number base 107$107$, since its covering set is {3,5,229}$\{3,5,229\}$, but 1071+122${107}^1+122$ is exactly 229$229$ and the (conjectured) smallest dual Sierpinski number base 107$107$ is 362$362$.

For b=2$b=2$, since the (conjectured) smallest Sierpinski number 78557$78557$ and the (conjectured) smallest Riesel number 509203$509203$ are coprime to 2$2$, they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.

For b=3$b=3$, since the (conjectured) smallest Sierpinski number 125050976086$125050976086$ and the (conjectured) smallest Riesel number 63064644938$63064644938$ are coprime to 3$3$, they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.

For b=4$b=4$, since the (conjectured) smallest Sierpinski number 66741$66741$ and the (conjectured) smallest Riesel number 39939$39939$ are coprime to 4$4$, they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.

For b=5$b=5$, since the (conjectured) smallest Sierpinski number 159986$159986$ and the (conjectured) smallest Riesel number 346802$346802$ are coprime to 5$5$, they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.

For b=6$b=6$, since neither the (conjectured) smallest Sierpinski number 174308$174308$ nor the (conjectured) smallest Riesel number 84687$84687$ are coprime to 6$6$, I continue to search and found the (conjectured) smallest dual Sierpinski number 282001$282001$ and the (conjectured) smallest dual Riesel number 333845$333845$.

For b=7$b=7$, since the (conjectured) smallest Sierpinski number 1112646039348$1112646039348$ and the (conjectured) smallest Riesel number 408034255082$408034255082$ are coprime to 7$7$, they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.

For b=8$b=8$, since the (conjectured) smallest Sierpinski number 47$47$ is coprime to 8$8$, it is also the (conjectured) smallest dual Sierpinski number, but the (conjectured) smallest Riesel number 14$14$ is not coprime to 8$8$, I continue to search and found the (conjectured) smallest dual Riesel number 209$209$.

For b=9$b=9$, since the (conjectured) smallest Sierpinski number 2344$2344$ and the (conjectured) smallest Riesel number 74$74$ are coprime to 9$9$, however, 92−74=7$9^2-74=7$ is prime and hence 74$74$ cannot be dual Riesel number, I continue to search and found the (conjectured) smallest dual Riesel number 18404$18404$, however, 2344$2344$ is really the (conjectured) smallest dual Sierpinski number.

For b=10$b=10$, since neither the (conjectured) smallest Sierpinski number 9175$9175$ nor the (conjectured) smallest Riesel number 10176$10176$ are coprime to 10$10$, I continue to search and found the (conjectured) smallest dual Sierpinski number 9351$9351$ and the (conjectured) smallest dual Riesel number 17601$17601$.

For b=11$b=11$, since the (conjectured) smallest Sierpinski number 1490$1490$ and the (conjectured) smallest Riesel number 862$862$ are coprime to 11$11$, they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.

For b=12$b=12$, since the (conjectured) smallest Sierpinski number 521$521$ is coprime to 12$12$, it is also the (conjectured) smallest dual Sierpinski number, but the (conjectured) smallest Riesel number 376$376$ is not coprime to 12$12$, I continue to search and found the (conjectured) smallest dual Riesel number 2261$2261$.

For b=13$b=13$, since the (conjectured) smallest Sierpinski number 132$132$ and the (conjectured) smallest Riesel number 302$302$ are coprime to 13$13$, they are also the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number, respectively.

For b=14$b=14$, since neither the (conjectured) smallest Sierpinski number 4$4$ nor the (conjectured) smallest Riesel number 4$4$ are coprime to 14$14$, I continue to search and found the (conjectured) smallest dual Sierpinski number 11$11$ and the (conjectured) smallest dual Riesel number 19$19$ (since 141−11=3${14}^1-11=3$ is prime, 11$11$ cannot be dual Riesel number).

For b=15$b=15$, neither the (conjectured) smallest Sierpinski number 91218919470156$91218919470156$ nor the (conjectured) smallest Riesel number 36370321851498$36370321851498$ are coprime to 15$15$, and they are too large to search. My question is:

What are the (conjectured) smallest dual Sierpinski number and the (conjectured) smallest dual Riesel number in base b=15$b=15$?