A268061 -id:A268061 - cp's OEIS frontend

A269544 Numbers n such that 7*8^n + 1 is prime.

2, 40, 58, 60, 130, 144, 752, 7462, 18162, 69028, 187272, 268178, 270410, 497284, 713304, 722600, 1005254

Offset: 1

Robert Price, Feb 29 2016

a(n) is even because 7*8^(2*k+1) + 1 is divisible by 3.

Select[Range[0, 200000,2], PrimeQ[7*8^# + 1] &]

is(n)=ispseudoprime(7*8^n+1) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A032353(n)/3 where this is an integer.

A272057 Numbers n such that 3*4^n - 1 is prime.

Original entry on oeis.org

1, 2, 3, 9, 17, 19, 32, 38, 47, 103, 108, 153, 162, 229, 235, 637, 1638, 2102, 2567, 6338, 7449, 12845, 20814, 40165, 61815, 77965, 117380, 207420, 351019, 496350, 600523, 1156367, 2117707, 5742009, 5865925, 5947859

Offset: 1

Tim Johannes Ohrtmann, Apr 19 2016

These are Williams primes to base 3.

Half of the even terms of A002235.

H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A002235, A000043, A003307, A046865, A079906, A046866, A268061, A268356, A056725, A046867, A079907.

Select[Range[0, 100000], PrimeQ[3*4^# - 1] &]

for(n=1,10000, if(isprime(3*4^n-1), print1(n,", ")))

a(25) corrected and a(33)-a(36) added by Giovanni Resta, Apr 19 2016, using data from A002235.

A297348 Numbers k such that 12*13^k - 1 is prime.

Original entry on oeis.org

0, 2, 7, 11, 36, 164, 216, 302, 311, 455, 738, 1107, 2244, 3326, 4878, 8067, 46466

Offset: 1

Tim Johannes Ohrtmann, Feb 15 2018

Williams primes to base 12.

Steven Harvey, Williams primes
H. C. Williams, The primality of certain integers of the form 2Ar^n - 1, Acta Arith. 39 (1981), 7-17.

Cf. A003307, A272057, A046865, A079906, A046866, A268061, A268356, A056725, A046867, A079907, A273523.

[n: n in [0..10000] |IsPrime(12*13^n-1)];

Select[Range[0, 10000], PrimeQ[12*13^n-1] &]

for(n=0, 10000, if(isprime(12*13^n-1), print1(n, ", ")))

a(16) from Michael S. Branicky, Sep 14 2022

a(17) from Michael S. Branicky, Nov 12 2024

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

Original entry on oeis.org

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

Eric Chen, Jun 04 2018

nonn

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.

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

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)).

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

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A269544 Numbers n such that 7*8^n + 1 is prime.

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A272057 Numbers n such that 3*4^n - 1 is prime.

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A297348 Numbers k such that 12*13^k - 1 is prime.

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

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PARI