A051783 -id:A051783 - cp's OEIS frontend

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

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)))

A089213 Primes p such that either 3^p-2 or 3^p+2 is prime or both are.

Original entry on oeis.org

2, 3, 5, 37, 41, 139, 317, 541, 2521

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net), Dec 20 2003

At p = 2, 3, 139, 3^p + 2 is prime, while at p = 2, 5, 37, 41, 317, 541, 2521 3^p - 2 is prime.

a(10) > 2*10^5. - Robert Price, Nov 20 2013

			2 is in the sequence because both 3^2 - 2 = 7 and 3^2 + 2 = 11 are primes.
3 is in the sequence because 3^3 + 2 = 29 is a prime (though 3^3 - 2 = 25 = 5^2).
5 is in the sequence because 3^5 - 2 = 241 is a prime (though 3^5 + 2 = 245 = 5 * 7^2).

Cf. A014224, A051783.

Select[Prime[Range[100]], PrimeQ[3^# - 2] || PrimeQ[3^# + 2] &] (* Alonso del Arte, Nov 20 2013 *)
Select[Prime[Range[400]],AnyTrue[3^#+{2,-2},PrimeQ]&] (* Harvey P. Dale, Aug 03 2025 *)

Edited by Zak Seidov, Aug 08 2006

Definition clarified by Harvey P. Dale, Aug 03 2025

A108259 Consider primes p and q such that p = 3^k + 2 and q = 3^(k+1) + 2 for some k; sequence gives values of p.

Original entry on oeis.org

3, 5, 11, 29, 4782971

Offset: 1

Cino Hilliard, Jun 29 2005

nonn

a(6) > 3^1400000 + 2, if it exists (cf. A051783). - Amiram Eldar, Jul 07 2024

			3^1 + 2 = 5, 3^2 + 2 = 11.

Cf. A051783.

3^#+2&/@Select[Range[0,20],AllTrue[{3^#+2,3^(#+1)+2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 01 2015 *)

g(m,n,b) = { for(x=0,n, y=m+b^x+b%2; z=m+b^(x+1)+b%2; if(isprime(y)&isprime(z),print1(p",") ) ) }

Offset corrected by Amiram Eldar, Jul 07 2024

A167675 Least prime p such that p-2 has n divisors, or 0 if no such prime exists.

Original entry on oeis.org

3, 5, 11, 17, 83, 47, 0, 107, 227, 569, 59051, 317, 0, 9479, 2027, 947, 0, 2207, 0, 2837, 88211, 295247, 0, 3467, 50627, 9034499, 11027, 47387, 0, 14177, 0, 15017, 1476227, 215233607, 455627, 17327, 150094635296999123, 15884240051, 89813531, 36857, 0

Offset: 1

T. D. Noe, Nov 09 2009

nonn

This sequence is the idea of Alonso Del Arte. For n>2, a(n) is conjectured to be the smallest number that is orderly (see A167408) for n-1 values of k. For example, 11 is orderly for k=3 and 9. See A056899 for other primes p that are orderly for two k. It is a conjecture because it is not known whether there are composite numbers that are orderly for more than one value of k.

The terms a(n) for prime n are 0 except when 3^(n-1)+2 is prime. Using A051783, we find the exceptional primes to be n=2, 3, 5, 11, 37, 127, 6959.... For these n, a(n) = 3^(n-1)+2. For any n, it is easy to use the factorization of n to find the forms of numbers that have n divisors. For example, for n=38=2*19, we know that the prime must have the form 2+q*r^18 with q and r prime. The smallest such prime is 2+41*3^18.

Cf. A066814 (smallest prime p such that p-1 has n divisors)

nn=25; t=Table[0,{nn}]; Do[p=Prime[n]; k=DivisorSigma[0,p-2]; If[k<=nn && t[[k]]==0, t[[k]]=p], {n,2,10^6}]; t

A248547 Numbers n such that 75^n+2 is prime.

Original entry on oeis.org

0, 12, 28, 53, 65, 504, 1967, 6915

Offset: 1

Vaclav Kotesovec, Oct 08 2014

Dedicated to N. J. A. Sloane for his 75th birthday!
Next term, if it exists, is greater than 10000.
Next term, if it exists, is greater than 50000. - Michael S. Branicky, May 21 2025

Cf. A051783, A087885, A090649, A109076, A138048, A113480, A138049, A138050.

[n: n in [0..200] | IsPrime(75^n+2)]; // Vincenzo Librandi, Oct 08 2014

Select[Range[0, 1000], PrimeQ[75^# + 2] &]

is(n)=ispseudoprime(75^n+2) \\ Charles R Greathouse IV, Jun 13 2017

A337837 Numbers k such that Omega(3^k - 2) = Omega(3^k + 2) where Omega is A001222.

Original entry on oeis.org

2, 4, 12, 18, 20, 28, 30, 31, 34, 35, 38, 44, 45, 49, 50, 58, 60, 75, 79, 97, 100, 103, 111, 113, 118, 120, 135, 141, 153, 154, 156, 166, 168, 171, 178, 181, 204, 219, 220, 239, 245, 247, 254, 260, 267, 269, 280, 286, 298, 307, 313

Offset: 1

Zak Seidov, Sep 25 2020

nonn

The corresponding values of Omega: 1, 1, 2, 3, 2, 2, 3, 3, 3, 3, 4, 2, 3, 3, 3, 3, 2, 6, 5, 4, 3, 4, 4, 4, 2, 4, 3, 3, 7, 4, 2, 4, 4, 4, 4, 5, 5, 5, 3, 5, 5, 6, 5, 6, 4, 5, 4, 5, 7, 6, 8.

			2 is a term since Omega(3^2 - 2) = Omega(7) = 1, and Omega(3^2 + 2) = Omega(11) = 1.

Cf. A001222, A014224, A051783, A058481, A168607.

Select[Range[200],PrimeOmega[3^#-2] == PrimeOmega[3^#+2]&]

for (k = 1, 200, if ((m = bigomega (3^k - 2)) == bigomega (3^k + 2), print (k ", " m ", ")))

a(36)-a(51) from Amiram Eldar, Sep 25 2020

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

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A089213 Primes p such that either 3^p-2 or 3^p+2 is prime or both are.

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A108259 Consider primes p and q such that p = 3^k + 2 and q = 3^(k+1) + 2 for some k; sequence gives values of p.

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A167675 Least prime p such that p-2 has n divisors, or 0 if no such prime exists.

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A248547 Numbers n such that 75^n+2 is prime.

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Magma

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A337837 Numbers k such that Omega(3^k - 2) = Omega(3^k + 2) where Omega is A001222.

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