A002253 -id:A002253 - cp's OEIS frontend

A268659 Numbers n such that 3*2^n + 1 is a prime factor of a generalized Fermat number 10^(2^m) + 1 for some m.

Original entry on oeis.org

209, 44685, 157169, 303093, 362765, 916773, 2145353

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=10)
OEIS Wiki, Generalized Fermat numbers

Cf. A080176, A268657, A268658, A204620, A268660, A268661, A268662, A268663, A226366, A268664. Subsequence of A002253.

A066466 Numbers having just one anti-divisor.

Original entry on oeis.org

3, 4, 6, 96, 393216

Offset: 1

Robert G. Wilson v, Jan 02 2002

See A066272 for definition of anti-divisor.
Jon Perry calls these anti-primes.
A066272(a(n)) = 1.
From Max Alekseyev, Jul 23 2007; updated Jun 25 2025: (Start)
Except for a(2) = 4, the terms of A066466 have form 2^k*p where p is odd prime and both 2^(k+1)*p-1, 2^(k+1)*p+1 are prime (i.e., twin primes). In other words, this sequence, omitting 4, is a subsequence of A040040 containing elements of the form 2^k*p with prime p.
Furthermore, since 2^(k+1)*p-1, 2^(k+1)*p+1 must equal -1 and +1 modulo 3, the number 2^(k+1)*p must be 0 modulo 3, implying that p=3. Therefore every term, except 4, must be of the form 3*2^k such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. In other words, k+1 belongs to the intersection of A002253 and A002235.
According to Ballinger and Keller's lists, there are no other such k up to 22*10^6. Therefore a(6) (if it exists) is greater than 3*2^(22*10^6) ~= 10^6622660. (End)
From Daniel Forgues, Nov 23 2009: (Start)
The 2 last known anti-primes seem to relate to the Fermat primes (coincidence?):
96 = 3 * 2^5 = 3 * 2^F_1 = 3 * 2^[2^(2^1) + 1] and
393216 = 3 * 2^17 = 3 * 2^F_2 = 3 * 2^[2^(2^2) + 1],
where F_k is the k-th Fermat prime. (End)

Ray Ballinger and Wilfrid Keller, List of primes k·2^n + 1 for k < 300
Wilfrid Keller, List of primes k·2^n - 1 for k < 300
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A000215, A002253, A002235, A066272, A019434.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n & ]; Select[ Range[10^5], Length[ antid[ # ]] == 1 & ]

Edited by Max Alekseyev, Oct 13 2009

A175172 Primes p such that 3*2^p+1 is also prime.

Original entry on oeis.org

2, 5, 41, 353

Offset: 1

Vincenzo Librandi, Mar 09 2010

If it exists, a(5) > 250000. - Hugo Pfoertner, Apr 12 2025

			For p=2, 3*2^2+1=13; p=5. 3*2^5+1=97; p=41, 3*2^41+1=6597069766657.

Cf. A000040, A002253.

[p: p in PrimesUpTo(4000) | IsPrime(3*2^p+1)];

select(p->isprime(p) and isprime(3*2^p+1),[$0..5000]); # Muniru A Asiru, Dec 19 2018

Select[Prime[Range[1000]], PrimeQ[3 2^# + 1] &] (* Vincenzo Librandi, Dec 19 2018 *)

A000040 INTERSECT A002253. - R. J. Mathar, Jan 04 2011

A112245 Numbers k such that 65537*2^k+1 is prime.

Original entry on oeis.org

287, 1695, 81359, 512895

Offset: 1

T. D. Noe, Aug 30 2005, Aug 26 2007

Note that 65537=2^16+1 is the largest known Fermat prime. These n yield provable primes. The primes are the smallest numbers in classes 303, 1711 and 81375 of the phi iteration (see A007755).

Jacques Molne found 512895. The corresponding provable prime is the smallest number in class 512911 of the Phi iteration.

Cf. A002253, A002254, A002259, A053345 (F*2^n+1 is prime, where F is a Fermat prime).

is(n)=ispseudoprime(65537*2^n+1) \\ Charles R Greathouse IV, Jun 13 2017

A263046 Smallest number k>2 such that k*2^n + 1 is a prime number.

Original entry on oeis.org

4, 3, 3, 5, 6, 3, 3, 5, 3, 15, 12, 6, 3, 5, 4, 5, 12, 6, 3, 11, 7, 11, 25, 20, 10, 5, 7, 15, 12, 6, 3, 35, 18, 9, 12, 6, 3, 15, 10, 5, 6, 3, 9, 9, 15, 35, 19, 27, 15, 14, 7, 14, 7, 20, 10, 5, 27, 29, 54, 27, 31, 36, 18, 9, 12, 6, 3, 9, 31, 23, 39, 39, 40, 20, 10, 5, 58

Offset: 0

Pierre CAMI, Oct 08 2015

nonn

If k = 2^j then 2^(n+j) + 1 is a Fermat prime.
a(n) = 3 if and only if 3*2^n + 1 is a prime; that is, n belongs to A002253. - Altug Alkan, Oct 08 2015
a(n+1) >= ceiling(a(n)/2). If a(n) is even then a(n+1) = a(n)/2. - Robert Israel, Oct 08 2015

			3*2^1 + 1 = 7 (prime), so a(1)=3:
3*2^2 + 1 = 13 (prime), so a(2)=3;
3*2^3 + 1 = 25 (composite), 4*2^3 + 1 = 33 (composite), 5*2^3 - 1 = 41 (prime), so a(3)=5.

Pierre CAMI, Table of n, a(n) for n = 0..10000

Cf. A247479, A262994.

f:= proc(n) local k;
    for k from 3 do if isprime(k*2^n+1) then return k fi od
  end proc:
seq(f(n),n=1..100); # Robert Israel, Oct 08 2015

Table[k = 3; While[! PrimeQ[k 2^n + 1], k++]; k, {n, 76}] (* Michael De Vlieger, Oct 08 2015 *)

a(n) = {k=3; while (! isprime(k*2^n+1), k++); k;} \\ Michel Marcus, Oct 08 2015

A282943 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 7^(2^m) + 1 for some m.

Original entry on oeis.org

8, 12, 36, 276, 408, 2208, 2816, 3168, 3912, 42665, 44685, 59973, 709968, 916773, 1832496

Offset: 1

Arkadiusz Wesolowski, Feb 25 2017

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to "Factors of generalized Fermat numbers", Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
OEIS Wiki, Generalized Fermat numbers

Cf. A078304, A204620, A268657, A268658, A268659, A282944, A268660.

Subsequence of A002253.

SetDefaultRealField(RealField(400)); IsInteger := func; [n: n in [2..408] | IsPrime(k) and IsInteger(Log(2, Modorder(7, k))) where k is 3*2^n+1];

lst = {}; Do[p = 3*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[7, p]], AppendTo[lst, n]], {n, 3912}]; lst

A282944 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 11^(2^m) + 1 for some m.

Original entry on oeis.org

6, 30, 36, 66, 276, 353, 2816, 3189, 34350, 48150, 80190, 1832496, 2291610, 5082306, 10829346

Offset: 1

Arkadiusz Wesolowski, Feb 25 2017

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to "Factors of generalized Fermat numbers", Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
OEIS Wiki, Generalized Fermat numbers

Cf. A199592, A204620, A268657, A268658, A282943, A268659, A268660.

Subsequence of A002253.

SetDefaultRealField(RealField(350)); IsInteger := func; [n: n in [2..353] | IsPrime(k) and IsInteger(Log(2, Modorder(11, k))) where k is 3*2^n+1];

lst = {}; Do[p = 3*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[11, p]], AppendTo[lst, n]], {n, 3189}]; lst

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)))

A238739 Numbers n such that 2^n + 3 and 3*2^n + 1 are both prime.

Original entry on oeis.org

1, 2, 6, 12, 18, 30

Offset: 1

Juri-Stepan Gerasimov, Mar 04 2014

Intersection of A057732 and A002253. - Joerg Arndt, Mar 04 2014
By checking primality of 2^n+3 for values n in A002253, it follows a(7) > 7033641. - Giovanni Resta, Mar 08 2014
Exponents of second Fermat prime pairs. - Juri-Stepan Gerasimov, Mar 08 2014
From Juri-Stepan Gerasimov, Mar 04 2014: (Start)
If prime pair {2^n + (2k+1), (2k+1)*2^n + 1} is called a Fermat prime pair, then numbers n such that 2^n + (2k + 1) and (2k + 1)*2^n + 1 are both prime:
for k = 0: 0, 1, 2, 4, 8, 16, ...   the exponents first Fermat prime pairs;
for k = 1: 1, 2, 6, 12, 18, 30, ... the exponents second Fermat prime pairs;
for k = 2: 1, 3, ...                the exponents third Fermat prime pairs;
for k = 3: 2, 4, 6, 20, 174, ...    the exponents fourth Fermat prime pairs;
for k = 4: 1, 2, 3, 6, 7, ...       the exponents fifth Fermat prime pairs;
for k = 5: 1, 3, 5, 7, ...          the exponents sixth Fermat prime pairs;
for k = 6: 2, 8, 20, ...            the exponents seventh Fermat prime pairs;
for k = 7: 1, 2, 4, 10, 12, ...     the exponents eighth Fermat prime pairs;
for k = 8:
for k = 9: 6, ...                   the exponents tenth Fermat prime pairs;
for k = 10: 1, 4, 5, 7, 16, ...     the exponents eleventh Fermat prime pairs;
for k = 11:
for k = 12: 2, 4, 6, 10, 20, 22, ...the exponents thirteenth Fermat prime pairs;
for k = 13: 2, 4, 16, 40, 44, ...   the exponents fourteenth Fermat prime pairs;
for k = 14: 1, 3, 5, 27, 43, ...    the exponents fifteenth Fermat prime pairs.
Semiprimes of the form (2^m+2k+1)*((2k+1)*2^m+1): 4, 9, 25, 35, 77, 91, 209, 289, 319, 481, 527, 533, 901, 989, ...
(End)

			a(1) = 1 because 2^1 + 3 = 5 and 3*2^1 + 1 = 7 are both prime,
a(2) = 2 because 2^2 + 3 = 7 and 3^2^2 + 1 = 13 are both prime,
a(3) = 6 because 2^6 + 3 = 67 and 3*2^6 + 1 = 193 are both prime.

Cf. A002253, A019434, A039687, A057732, A057733, A238554, A239038, A267984.

[n: n in [0..30] | IsPrime(2^n+3) and IsPrime(3*2^n+1)]; // Arkadiusz Wesolowski, Jan 23 2016

Select[Range[30],AllTrue[{2^#+3,3*2^#+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 08 2015 *)

isok(n) = isprime(2^n + 3) && isprime(3*2^n + 1); \\ Michel Marcus, Mar 04 2014

A281483 Numbers k such that 32771*2^k + 1 is prime.

Original entry on oeis.org

1, 13, 19, 29, 37, 45, 51, 61, 63, 65, 69, 117, 171, 181, 199, 201, 217, 221, 265, 337, 627, 631, 881, 1035, 1507, 1525, 1627, 1641, 2037, 3175, 4639, 6445, 21537, 29801, 30521, 30917, 37877, 49725, 50877, 57537, 61337, 118141, 125169, 200961, 204117, 283445, 395125, 829489

Offset: 1

Pierre CAMI, Jan 22 2017

nonn

a(48) = 829489, 1438879 is such that 32771*2^1438879 + 1 is prime with 433151 digits, 829489 < a(49) <= 1438879.

Cf. A002253.

Select[Range@ 3000, PrimeQ[32771*2^# + 1] &] (* Michael De Vlieger, Jan 23 2017 *)

list(limit)=my(i=1); while(iAnders Hellström, Feb 04 2017

32771*2^$a+1
$a: from 1 to 1000001 step 2

cp's OEIS Frontend

A268659 Numbers n such that 3*2^n + 1 is a prime factor of a generalized Fermat number 10^(2^m) + 1 for some m.

Views

Author

Keywords

References

Links

Crossrefs

A066466 Numbers having just one anti-divisor.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A175172 Primes p such that 3*2^p+1 is also prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A112245 Numbers k such that 65537*2^k+1 is prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A263046 Smallest number k>2 such that k*2^n + 1 is a prime number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A282943 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 7^(2^m) + 1 for some m.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A282944 Numbers k such that 3*2^k + 1 is a prime factor of a generalized Fermat number 11^(2^m) + 1 for some m.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A238739 Numbers n such that 2^n + 3 and 3*2^n + 1 are both prime.

Views

Author