A087139 -id:A087139 - cp's OEIS frontend

A087126 Primes of the form p^k - p^(k-1) + 1 for some prime p and integer k > 1.

3, 5, 7, 17, 19, 43, 101, 157, 163, 257, 487, 1459, 2029, 4423, 6163, 14407, 19183, 22651, 23549, 26407, 37057, 39367, 62501, 65537, 77659, 113233, 121453, 143263, 208393, 292141, 342733, 375157, 412807, 527803, 564899, 590593, 697049, 843643

Offset: 1

T. D. Noe, Aug 15 2003

nonn

It is usually the case that, for prime p and k > 1, the first time the totient function phi(n) has value p^k - p^(k-1) is for n = p^k. However, this is not true when p^k - p^(k-1) + 1 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Totient Function
Eric Weisstein's World of Mathematics, Totient Valence Function

Cf. A002383 (primes of the form n^2 + n + 1, which is the same as n^2 - n + 1).

Cf. A019434 (Fermat primes), A003306 (2*3^n + 1 is prime), A056799 (8*9^n + 1 is prime), A056797 (9*10^n + 1 is prime), A087139 (least k such that p^k - p^(k-1) + 1 is prime for p = prime(n)).

lst={}; maxNum=10^6; n=1; While[p=Prime[n]; p^2-p+1

A122396 Least k>1 such that p^k - p^(k-1) - 1 is prime for p = prime(n).

Original entry on oeis.org

3, 2, 2, 2, 2, 3, 2, 7, 56, 2, 2, 8, 8, 8, 2, 4, 4, 2, 2, 2, 9, 3, 21496, 26, 2, 2, 4, 38, 7, 286644, 2, 2, 26, 2, 2, 4, 4, 15, 4, 24, 16, 2, 264, 4, 2, 3, 24, 3, 516, 6

Offset: 1

T. D. Noe, Aug 31 2006

Does a(n) always exist? Note that k cannot be 5, 11, 17,... (i.e., k=5 mod 6) because then p^2 - p + 1 divides p^k - p^(k-1) - 1.
From Richard N. Smith, Jul 15 2019: (Start)
The link has the primes 82*83^21495-1 = 83^21496-83^21495-1 and 112*113^286643-1 = 113^286644-113^286643-1, thus a(23)=21496 and a(30)=286644.
a(51) > 250000, since 232*233^k-1 is composite for all k<=250000, see link.
a(52) - a(61) = {4, 2, 80, 14, 76, 2, 90, 6, 80, 769}, a(62) > 200000. (End)

Steven Harvey, Williams primes

Cf. A087139, A122395.

lst={}; Do[p=Prime[n]; k=2; While[m=p^k-p^(k-1)-1; !PrimeQ[m], k++ ]; AppendTo[lst,k], {n,22}]; lst

a(n)=for(k=2, 10^6, if(ispseudoprime(prime(n)^k - prime(n)^(k-1) - 1), return(k))) \\ Richard N. Smith, Jul 15 2019

a(23)-a(50) from Richard N. Smith, Jul 15 2019, using Steven Harvey's table.

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)))

cp's OEIS Frontend

A087126 Primes of the form p^k - p^(k-1) + 1 for some prime p and integer k > 1.

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A122396 Least k>1 such that p^k - p^(k-1) - 1 is prime for p = prime(n).

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

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PARI