A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1

Offset: 2

Author

Eric Chen, Jun 04 2018

Keywords

• nonn

Comments

a(prime(j)) + 1 = A087139(j).

a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.

a(251) > 73000, see A087139.

Links

• Eric Chen, Table of n, a(n) for n = 2..122
• Gary Barnes, Sierpinski conjectures and proofs
• Eric Chen, Table n, a(n) for n = 2..360 status

Crossrefs

For the numbers k such that these forms are prime:

a1(b): numbers k such that (b-1)*b^k-1 is prime

a2(b): numbers k such that (b-1)*b^k+1 is prime

a3(b): numbers k such that (b+1)*b^k-1 is prime

a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))

a5(b): numbers k such that b^k-(b-1) is prime

a6(b): numbers k such that b^k+(b-1) is prime

a7(b): numbers k such that b^k-(b+1) is prime

a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).

Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):

.

b a1(b) a2(b) a3(b) a4(b) a5(b) a6(b) a7(b) a8(b)

--------------------------------------------------------------------

2 A000043 ------- A002235 A002253 A000043 ------- A050414 A057732

3 A003307 A003306 A005540 A005537 A014224 A051783 A058959 A058958

4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx

5 A046865 A204322 A257790 A143279 A059613 A124621 A165701 A089142

6 A079906 A247260 ------- ------- A059614 A145106 A217352 A217351

7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 xxxxxxx

8 A268061 A269544 ------- ------- A217380 A217381 A217383 A217382

9 A268356 A056799 ------- ------- A177093 A217385 A217493 A217492

10 A056725 A056797 A111391 xxxxxxx A095714 A088275 A092767 xxxxxxx

11 A046867 A057462 ------- ------- ------- ------- ------- -------

12 A079907 A251259 ------- ------- ------- A137654 ------- -------

13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx

14 A273523 ------- ------- ------- ------- ------- ------- -------

15 ------- ------- ------- ------- ------- ------- ------- -------

16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx

Cf. (smallest k such that these forms are prime) A122396 (a1(b)+1 for prime b), A087139 (a2(b)+1 for prime b), A113516 (a5(b)), A076845 (a6(b)), A178250 (a7(b)).

Programs

PARI

a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))