A057732 -id:A057732 - cp's OEIS frontend

A153503 Primes p such that 2^(p-1)+3 is prime.

2, 3, 5, 7, 13, 17, 19, 29, 31, 229, 2371, 4003, 33029, 55457, 58313, 205963, 1875553

Offset: 1

Vincenzo Librandi, Dec 28 2008

A prime p is in the sequence if and only if p-1 is in A057732.

			For p = 2, 2^(p-1)+3 = 5 is prime.
For p = 17, 2^(p-1)+3 = 65539 is prime.
For p = 31, 2^(p-1)+3 = 1073741827 is prime.

Cf. A057732 (numbers k such that 2^k + 3 is prime), A057736 (primes p such that 2^p + 3 is prime), A000043 (primes p such that 2^p - 1 is prime).

[p: p in PrimesUpTo(2000) | IsPrime(2^(p-1) + 3)]; // Vincenzo Librandi, Jun 09 2015

Select[Prime[Range[3000]], PrimeQ[2^(# - 1) + 3] &] (* Vincenzo Librandi, Jun 09 2015 *)

Edited and a(13)-a(15) (based on A057732) added by Klaus Brockhaus, Jan 06 2009
a(16) from Vincenzo Librandi, Jun 09 2015
a(17) from Amiram Eldar, Aug 01 2024

A165991 Numbers n such that 3 + 2^n is not prime.

Original entry on oeis.org

0, 5, 8, 9, 10, 11, 13, 14, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87

Offset: 1

Vincenzo Librandi, Oct 03 2009

Complement of A057732.

			For n=0, 3+2^0=4 is not prime. n=5, 3+2^5=35=5*7, not prime. n=8, 3+2^8=259=7*37 not prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

[n: n in [0..100] |not IsPrime(3 + 2^n )]; // Vincenzo Librandi, Oct 15 2012

Select[Range[0, 100], !PrimeQ[3 + 2^#] &] (* Vincenzo Librandi, Oct 15 2012 *)

Entries checked - R. J. Mathar, Oct 05 2009

A229065 Numbers of the form 2^(p-1)+3, where p is prime.

Original entry on oeis.org

5, 7, 19, 67, 1027, 4099, 65539, 262147, 4194307, 268435459, 1073741827, 68719476739, 1099511627779, 4398046511107, 70368744177667, 4503599627370499, 288230376151711747, 1152921504606846979, 73786976294838206467, 1180591620717411303427, 4722366482869645213699

Offset: 1

Vincenzo Librandi, Sep 17 2013

nonn

Primes in the sequence: 5, 7, 19, 67, 4099, 65539, 262147, 268435459, 1073741827, ...

On the other hand, for example, 2^(p-1) + 3 is composite when p == 11 (mod 12) or p == 5 (mod 18), with p>5; or when p is of the form 2*h^2+2*h*(k+2)+3*k, with k>0 and h>1.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A153503 (associated primes p), A098828, A057732, A057736.

Magma
```
[2^(p-1)+3:  p in PrimesUpTo(80)];
```
Mathematica
```
Table[2^(Prime[n] - 1) + 3, {n, 25}]
```

A245476 Least number k > 1 such that k^n + k + 1 is prime, or 0 if no such number exists.

Original entry on oeis.org

2, 2, 2, 2, 0, 2, 2, 0, 3, 3, 0, 2, 5, 0, 2, 2, 0, 2, 8, 0, 6, 3, 0, 6, 15, 0, 6, 2, 0, 2, 23, 0, 23, 56, 0, 15, 114, 0, 14, 11, 0, 3, 14, 0, 29, 110, 0, 21, 9, 0, 53, 59, 0, 6, 2, 0, 3, 29, 0, 71, 21, 0, 146, 17, 0, 35, 2, 0, 9, 6, 0, 77, 41, 0, 27, 176, 0, 153, 21, 0, 39, 32, 0, 2, 314, 0, 3, 5, 0, 66, 44, 0, 234

Offset: 1

Derek Orr, Jul 23 2014

nonn

Except for a(2), a(n) = 0 if n == 2 mod 3 (A016789).
It appears that this is an "if and only if".
a(n) = 2 if and only if n is in A057732.
Many terms in the linked table correspond to probable primes. If n == 2 mod 3 then k^2+k+1 divides k^n+k+1. This is why a(n) = 0 if n > 2 and n == 2 mod 3. - Jens Kruse Andersen, Jul 28 2014

			2^9 + 2 + 1 = 515 is not prime. 3^9 + 3 + 1 = 19687 is prime. Thus a(9) = 3.

Robert Israel and Jens Kruse Andersen, Table of n, a(n) for n = 1..1000 (first 640 terms from Robert Israel)

Cf. A127599, A016789, A057732.

Cf. Numbers n such that n^s + n + 1 is prime: A005097 (s = 1), A002384 (s = 2), A049407 (s = 3), A049408 (s = 4), A075723 (s = 6), A075722 (s = 7), A075720 (s = 9), A075719 (s = 10), A075718 (s = 12), A075717 (s = 13), A075716 (s = 15), A075715 (s = 16), A075714 (s = 18), A075713 (s = 19).

f:= proc(n) local k;
   if n mod 3 = 2 and n > 2 then return 0 fi;
   for k from 2 to 10^6 do
      if isprime(k^n+k+1) then return k fi
   od:
  error("no solution found for n = %1",n);
end proc:
seq(f(n),n=1..100); # Robert Israel, Jul 27 2014

a(n) = if(n>2&&n==Mod(2, 3), return(0)); k=2; while(!ispseudoprime(k^n+k+1), k++); k
vector(150, n, a(n)) \\ Derek Orr with corrections and improvements from Colin Barker, Jul 23 2014

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)))

A329102 Numbers k such that both k^2 + 3 and 2^k + 3 are primes.

Original entry on oeis.org

2, 4, 28, 784

Offset: 1

Alex Ratushnyak, Nov 04 2019

Intersection of A057732 and A049422.
a(5) > 20000. - Tyler NeSmith, May 15 2021
a(5) > 2205444, using A057732. - Michael S. Branicky, Feb 16 2024

Cf. A000040, A057732, A049422, A329103.

A331487 Primes p such that exactly one of 2^(p+1) - 3 and 2^(p+1) + 3 is a prime.

Original entry on oeis.org

13, 17, 19, 23, 29, 83, 149, 173, 227, 389, 1109, 4001, 35753, 36551, 363119, 702193

Offset: 1

Juri-Stepan Gerasimov, Jan 18 2020

Primes p such that exactly one of k*2^p - 2*k + 1 and k*2^p + 2*k - 1 is a prime:
k = 1: odd terms in A000043;
k = 2: this sequence;
k = 3: 5, 13, 19, 29, 31, 109, 139, 271, 379, 1553, ...
k = 4: 2, 37, ...
k = 5: 3, 5, 7, 17, 19, 23, 41, 61, 67, 151, 157, 313, 4111, 6337, ...
k = 6: 2, 5, 7, 11, 19, 29, 149, 191, 373, 449, 983, 1667, 1973, ...
k = 7: 2, 3, 5, 7, 11, 13, 29, 43, 61, 97, 109, 127, 131, 239, 461, 1153, ...
k = 8: 3, 11, 19, 23, 29, 37, 43, 97, 193, 307, 617, 1847, ...
k = 9: 3, 5, 23, 41, 61, 71, 97, 131, 157, 863, 3119, ...
k = 10: 2, 3, 13, ...
...

			13 is in this sequence because 2^(13+1) - 3 = 16381 (prime) and 2^(13+1) + 3 = 16387 (composite number).

Cf. A000043, A050414, A057732, A174269.

[p: p in PrimesUpTo(1000) | not (#[k: k in [2] | IsPrime(k*2^p-2*k+1)]) eq (#[k: k in [2] | IsPrime(k*2^p+2*k-1)])];

Select[Range[400], PrimeQ[#] && Xor @@ PrimeQ[2^(# + 1) + {-3, 3}] &] (* Amiram Eldar, Jan 19 2020 *)

isok(p) = isprime(2*2^p-3) + isprime(2*2^p+3) == 1;
forprime(p=2, 500, if(isok(p), print1(p, ", "))); \\ Jinyuan Wang, Jan 19 2020

a(12)-a(16) added using A050414 and A057732 by Jinyuan Wang, May 15 2020

A071781 Primes p with p-2^e and p+2^e prime for some exponent e.

Original entry on oeis.org

5, 7, 11, 67, 32771

Offset: 1

Rick L. Shepherd, Jun 05 2002

For each n, p-2^e,p,p+2^e is thus an arithmetic progression of primes with difference 2^e. Note that for each n=1,2,3,4,5, only one such e exists and p-2^e=3. There are no other terms up to 20000000.

For all terms, p-2^e must, in fact, be 3 (as one of p-2^e, p and p+2^e is divisible by 3). Each corresponding arithmetic progression of primes has length 3 (p+2^(e+1) is also divisible by 3). Any additional term is too large to include here. Equivalently, this sequence is primes of the form 3+2^e such that 3+2^(e+1) is also prime; i.e., 3+2^A057732(k) is a term iff A057732(k+1) = A057732(k) + 1. Thus much more efficient than the PARI program below is to extend A057732 and examine its terms. - Rick L. Shepherd, Jun 20 2008

			67 is a term because 67 is prime and there exists e=6 such that both 67-2^6=67-64=3 and 67+2^6=67+64=131 are primes. 32771 is a term because 32771 is prime and there exists e=15 such that both 32771-2^15=32771-32768=3 and 32771+2^15=32771+32768=65539 are primes. Thus 3,67,131 and 3,32771,65539 are two sequences of primes in arithmetic progression with differences 2^6 and 2^15, respectively.

Cf. A056206, A056208.

Cf. A057732.

for(p=5,20000000,if(isprime(p),e=1; while(p-2^e>1,if(isprime(p-2^e)&&isprime(p+2^e),print1(p,","); break,e++))))

A171131 Primes p such that sum of divisors of p-3 is prime.

Original entry on oeis.org

5, 7, 19, 67, 4099, 65539, 262147, 1073741827

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 04 2009

nonn

No further terms up to the 10 millionth prime. - Harvey P. Dale, Apr 30 2012

If the sum of divisors of a number k is a prime (i.e., k is in A023194), then k is a prime power. If p is prime and p-3 is a prime power, then p-3 is even, so p-3 is a power of 2. Since p-3 = 2^m then sigma(2^m) = 2^(m+1)-1 is a prime. Therefore, all the terms are primes of the form 2^m+3 where m+1 is a Mersenne exponent (A000043), i.e., m is in the intersection of A057732 and {A000043(n)-1}. So, m = 1, 2, 4, 6, 12, 16, 18, 30, and no other value <= A057732(58) = 2205444. Therefore, a(9) > 2^2205444, if it exists. - Amiram Eldar, Aug 01 2024

			5 is a term since it is a prime and sigma(5-3) = 3 is a prime.
7 is a term since it is a prime and sigma(7-3) = 7 is a prime.
19 is a term since it is a prime and sigma(19-3) = 31 is a prime.

Cf. A000043, A000203, A023194, A057732, A171130, A153503.

f[n_]:=Plus@@Divisors[n]; lst={};Do[p=Prime[n];If[PrimeQ[f[p-3]],AppendTo[lst,p]],{n,2*10!}];lst
Select[Prime[Range[10000000]],PrimeQ[DivisorSigma[1,#-3]]&] (* Harvey P. Dale, Apr 30 2012 *)

a(8)-a(10) from Vincenzo Librandi, Feb 04 2013

Two wrong terms removed by Amiram Eldar, Aug 01 2024

A174272 Exactly one of 2^n-3 and 2^n+3 is prime.

Original entry on oeis.org

1, 2, 5, 7, 9, 10, 14, 15, 16, 18, 20, 22, 24, 28, 29, 30, 55, 67, 84, 94, 116, 122, 150, 174, 213, 221, 228, 233, 266, 336, 390, 452, 545, 689, 694, 784, 850, 1110, 1704, 1736, 2008, 2139, 2191, 2321, 2367, 2370, 3237, 3954, 4002, 4060, 4062, 4552, 5547, 5630

Offset: 1

Juri-Stepan Gerasimov, Mar 14 2010

nonn

Numbers n which are in A050414 or in A057732 but not in both. [From R. J. Mathar, Mar 29 2010]

			a(1)=1 because 2^1-3=-1 is nonprime and 2^1+3=5 is prime.

Cf. A050414, A057732.

Corrected by Charles R Greathouse IV, Mar 20 2010

cp's OEIS Frontend

A153503 Primes p such that 2^(p-1)+3 is prime.

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Magma

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A165991 Numbers n such that 3 + 2^n is not prime.

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Magma

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A229065 Numbers of the form 2^(p-1)+3, where p is prime.

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Magma

Mathematica

A245476 Least number k > 1 such that k^n + k + 1 is prime, or 0 if no such number exists.

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Maple

PARI

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

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PARI

A329102 Numbers k such that both k^2 + 3 and 2^k + 3 are primes.

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A331487 Primes p such that exactly one of 2^(p+1) - 3 and 2^(p+1) + 3 is a prime.

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Magma

Mathematica

PARI

Extensions

A071781 Primes p with p-2^e and p+2^e prime for some exponent e.

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PARI