A000978 -id:A000978 - cp's OEIS frontend

A049883 Primes in the Jacobsthal sequence (A001045).

3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243

Offset: 1

Judson Neer

nonn

All terms, except a(2) = 5, are of the form (2^p + 1)/3 - the Wagstaff primes A000979 = {3, 11, 43, 683, 2731, 43691, 174763, ...}.
Indices of prime Jacobsthal numbers are listed in A107036 = {3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, ...}.
For n > 1, A107036(n) = A000978(n) (numbers m such that (2^m + 1)/3 is prime). - Alexander Adamchuk, Oct 10 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..23

Cf. A001045, A000978, A000979, A107036.

Select[Table[(2^n + (-1)^(n - 1))/3, {n, 200}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2011 *)

A123627 Smallest prime q such that (q^p+1)/(q+1) is prime, where p = prime(n); or 0 if no such prime q exists.

Original entry on oeis.org

0, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 19, 61, 2, 7, 839, 89, 2, 5, 409, 571, 2, 809, 227, 317, 2, 5, 79, 23, 4073, 2, 281, 89, 739, 1427, 727, 19, 19, 2, 281, 11, 2143, 2, 1013, 4259, 2, 661, 1879, 401, 5, 4099, 1579, 137, 43, 487, 307, 547, 1709, 43, 3, 463, 2161, 353, 443, 2

Offset: 1

Alexander Adamchuk, Oct 04 2006, Aug 05 2008

nonn

a(1) = 0 because such a prime does not exist, Mod[n^2+1,n+1] = 2 for n>1.
Corresponding primes (q^p+1)/(q+1), where prime q = a(n) and p = Prime[n], are listed in A123628[n] = {1,3,11,43,683,2731,43691,174763,2796203,402488219476647465854701,715827883,...}.
a(n) coincides with A103795[n] when A103795[n] is prime.
a(n) = 2 for n = PrimePi[A000978[k]] = {2,3,4,5,6,7,8,9,11,14,18,22,26,31,39,43,46,65,69,126,267,380,495,762,1285,1304,1364,1479,1697,4469,8135,9193,11065,11902,12923,13103,23396,23642,31850,...}.
Corresponding primes of the form (2^p + 1)/3 are the Wagstaff primes that are listed in A000979[n] = {3,11,43,683,2731,43691,174763,2796203,715827883,...}.

Robert Israel, Table of n, a(n) for n = 1..240

Cf. A123628, A103795, A123487, A123488, A000978, A000979.

f:= proc(n) local p,q;
  p:= ithprime(n);
  q:= 1;
  do
   q:= nextprime(q);
   if isprime((q^p+1)/(q+1)) then return q fi
  od
end proc:
f(1):= 0:
map(f, [$1..70]); # Robert Israel, Jul 31 2019

a(1) = 0, for n>1 Do[p=Prime[k]; n=1; q=Prime[n]; cp=(q^p+1)/(q+1); While[ !PrimeQ[cp], n=n+1; q=Prime[n]; cp=(q^p+1)/(q+1)]; Print[q], {k, 2, 61}]
Do[p=Prime[k]; n=1; q=Prime[n]; cp=(q^p+1)/(q+1); While[ !PrimeQ[cp], n=n+1; q=Prime[n]; cp=(q^p+1)/(q+1)]; Print[{k,q}], {k, 1, 134}]
spq[n_]:=Module[{p=Prime[n],q=2},While[!PrimeQ[(q^p+1)/(q+1)],q=NextPrime[ q]]; q]; Join[{0},Array[spq,70,2]] (* Harvey P. Dale, Mar 23 2019 *)

A123628(n) = (a(n)^prime(n) + 1) / (a(n) + 1).

A127962 Binary expansion of A000979(n).

Original entry on oeis.org

11, 1011, 101011, 1010101011, 101010101011, 1010101010101011, 101010101010101011, 1010101010101010101011, 101010101010101010101010101011, 101010101010101010101010101010101010101011

Offset: 1

Artur Jasinski, Feb 09 2007

Amiram Eldar, Table of n, a(n) for n = 1..20

Cf. A000978, A000979, A007088, A124400, A126614, A127936, A127955, A127956, A127957, A127958, A127961.

b[n_] := FromDigits[IntegerDigits[n, 2]]; b /@ Select[Table[(2^p + 1)/3, {p, Prime[Range[15]]}], PrimeQ] (* Amiram Eldar, Jul 23 2023 *)

from gmpy2 import divexact
from sympy import prime, isprime
A127962 = [int(bin(p)[2:]) for p in (divexact(2**prime(n)+1,3) for n in range(2,10**2)) if isprime(p)] # Chai Wah Wu, Sep 04 2014

a(n) = A007088(A000979(n)). - Amiram Eldar, Jul 23 2023

Edited by N. J. A. Sloane, Jun 11 2007

A229524 Numbers k such that (38^k + 1)/39 is prime.

Original entry on oeis.org

5, 167, 1063, 1597, 2749, 3373, 13691, 83891, 131591

Offset: 1

Robert Price, Sep 25 2013

All terms are primes. a(9) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145.

Do[ p=Prime[n]; If[ PrimeQ[ (38^p + 1)/39 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((38^n+1)/39) \\ Charles R Greathouse IV, Feb 17 2017

a(9)=131591 corresponds to a probable prime discovered by Paul Bourdelais, Jul 03 2018

A125956 Numbers k such that (2^k + 9^k)/11 is prime.

Original entry on oeis.org

3, 7, 127, 283, 883, 1523, 4001

Offset: 1

Alexander Adamchuk, Feb 06 2007

All terms are primes. Note that first 3 terms {3, 7, 127} are primes of the form 2^q - 1, where q = {2, 3, 7} is prime too. Corresponding primes of the form (2^k + 9^k)/11 are {67, 434827, ...}.

a(8) > 10^5. - Robert Price, Dec 23 2012

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime.
Cf. A057469 = numbers n such that (2^n + 3^n)/5 is prime.
Cf. A082387 = numbers n such that (2^n + 5^n)/7 is prime.

Do[p=Prime[n];f=(2^p+9^p)/11; If[PrimeQ[f], Print[{p, f}]], {n, 1, 100}]

is(n)=ispseudoprime((2^n+9^n)/11) \\ Charles R Greathouse IV, Feb 20 2017

2 more terms from Rick L. Shepherd, Feb 14 2007

a(7) from Robert Price, Dec 23 2012

A073936 Numbers k such that 2^k + 1 is the product of two distinct primes.

Original entry on oeis.org

5, 6, 7, 11, 12, 13, 17, 19, 20, 23, 28, 31, 32, 40, 43, 61, 64, 79, 92, 101, 104, 127, 128, 148, 167, 191, 199, 256, 313, 347, 356, 596, 692, 701, 1004, 1228, 1268, 1709, 2617, 3539, 3824, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239

Offset: 1

Labos Elemer, Aug 13 2002

nonn

Original name: "2^n + 1 is squarefree and has exactly 2 prime factors."
From Giuseppe Coppoletta, May 08 2017: (Start)
As 3 divides 2^a(n) + 1 for any odd term a(n), all odd terms are prime and exactly the Wagstaff primes (A000978), at the exclusion of 3 (which gives 2^3 + 1 = 3^2 not squarefree).
For the even terms, let a(n) = d * 2^j with d odd integer and j > 0. If d > 1, as (2^2^j)^q  + 1 divides 2^a(n) + 1 for any odd prime q dividing d, then d must be prime.
So the even terms are all given by the following two class:
a) (d = 1) a(n) = 2^j such that Fj is a semiprime Fermat number. Up to now, only j = 5, 6, 7, 8 are known to give a Fermat semiprime, giving the even terms 32, 64, 128 and 256. We are also assured that 2^j is not a term for j = 9..19 because Fj is not a semiprime for those value of j (see Wagstaf's link). F20 is the first composite Fermat number which could give another even term (it would be 2^20 = 1048576). However, it seems highly unlikely that other Fermat semiprimes could exist.
b) (d = p odd prime) a(n) = p * 2^j with j such that Fj is a Fermat prime and p a prime verifying ((Fj - 1)^p + 1)/Fj is a prime.
Exemplifying that, we have:
for j = 1 this gives only the even term a(2) = 2 * 3 = 6  (see Jack Brennen's result in ref),
for j = 2 we have all the terms of type 2^2 * A057182.
for j = 3 the even terms are of type 2^3 * A127317.
For j = 4 at least up to 200000, there is only the term a(41) = 2^4 * 239 = 3824 (see comment in A127317).
All terms after a(50) refer to probabilistic primality tests for 2^a(n) + 1  (see Caldwell's link for the list of the largest certified Wagstaff primes).
After a(56), from the above, the primes 267017, 269987, 374321, 986191, 4031399 and the even value 4101572 are also terms, but still remains the (remote) possibility of some gaps in between. In addition, 13347311 and 13372531 are also terms, but possibly much further in the numbering (see comments in A000978).
(End)
Intersection of A092559 and A066263. - Eric Chen, Jun 13 2018

			11 is a member because 1 + 2^11 = 2049 = 3 * 683.
9 is not a term because 1 + 2^9 = 513 = 3^3 * 19

Giuseppe Coppoletta, Table of n, a(n) for n = 1..56
AMS Books Online, Factorizations of b^n = +-1, b=2,3,5,6,7,10,11,12 Up to High Powers, Third Edition.
Arjen Bot, Factors for 2^n-1 and 2^n+1 for 1200 < n < 10000.
Jack Brennen, Primes of the form (4^p+1)/5^t, Seqfan (Mar 15 2017).
C. Caldwell's The Top Twenty Wagstaff primes.
Mersennewiki, Factorizations Of Cunningham Numbers C+(2,n) (tables).
Samuel S. Wagstaff, The Cunningham Project.

Cf. A000005, A000051, A046798, A092559, A000978. Different from A066263.

Do[ If[ Length[ Divisors[1 + 2^n]] == 4, Print[n]], {n, 1, 200}]
(* Second program: *)
Select[Range@ 200, DivisorSigma[0, 2^# + 1] == 4 &] (* Michael De Vlieger, May 09 2017 *)

[n for n in xsrange(3,200) if sigma(2^n+1,0)==4]
# Second program (faster):

v=[]; N=2000
for n in xsrange(4,N):
    j=valuation(n,2)
    if j<5:
        Fj=2^2^j+1; p=ZZ(n/2^j); q=ZZ((2^n+1)/Fj)
        if p.is_prime() and q.is_prime(proof=false): v.append(n)
    elif j<9 and n.is_power_of(2): v.append(n)
print(v) # Giuseppe Coppoletta, May 11 2017

Solutions to A000005[A000051(x)]=4 or A046798[x]=4

Edited by Robert G. Wilson v, Aug 19 2002
a(28)-a(51) by Giuseppe Coppoletta, May 02 2017
Name reworded by Jon E. Schoenfield, Jun 15 2018

A229663 Numbers n such that (40^n + 1)/41 is prime.

Original entry on oeis.org

53, 67, 1217, 5867, 6143, 11681, 29959

Offset: 1

Robert Price, Sep 27 2013

All terms are primes.

a(8) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524.

Do[ p=Prime[n]; If[ PrimeQ[ (40^p + 1)/41 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((40^n+1)/41) \\ Charles R Greathouse IV, Feb 17 2017

A230036 Numbers n such that (39^n + 1)/40 is prime.

Original entry on oeis.org

3, 13, 149, 15377

Offset: 1

Robert Price, Oct 05 2013

All terms are primes.

a(5) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A000978 (numbers n such that (2^n + 1)/3 is prime).

Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524.

Do[ p=Prime[n]; If[ PrimeQ[ (39^p + 1)/40 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((39^n+1)/40) \\ Charles R Greathouse IV, Feb 17 2017

A066263 Numbers k such that 2^k + 1 has just two distinct prime factors.

Original entry on oeis.org

5, 6, 7, 9, 10, 11, 12, 13, 17, 19, 20, 23, 28, 31, 32, 40, 43, 61, 64, 79, 92, 101, 104, 127, 128, 148, 167, 191, 199, 256, 313, 347, 356, 596, 692, 701, 1004, 1228, 1268, 1709, 2617, 3539, 3824, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369

Offset: 1

Benoit Cloitre, Dec 31 2001

nonn

From Giuseppe Coppoletta, May 16 2017: (Start)
All terms after a(52) refer to probabilistic primality tests for 2^a(n) + 1 (see Caldwell's link for the list of the largest certified Wagstaff primes). After a(58), 267017, 269987, 374321, 986191, 4031399 and 4101572 are also terms, but there still remains the remote possibility of some gaps in between. In addition, 13347311 and 13372531 are also terms, but possibly much farther in the numbering (see comments in A000978).
For the relation with Fermat numbers and for the primality of odd terms, see comments in A073936. The terms 9 and 10 give a value of 2^n + 1 which is not squarefree, so they are not in A073936. For the rest, the actually known terms of the two sequences coincide. In order to verify if any other term could be found hereafter that is not in A073936, all we have to do is to examine the terms for which 2^n + 1 is not squarefree. Considering that 3 divides 2^a(n) + 1 for any odd term a(n) and using Zsigmondy's and Mihăilescu-Catalan's theorems (see links), one can verify that any nonsquarefree term greater than 9 has to be of the form a(n) = 2^j * Fj, where Fj is the Fermat prime 2^2^j + 1. So basically we have to see if ((Fj-1)^Fj + 1)/(Fj)^2 is a prime or the power of a prime for any Fermat prime Fj. The case j = 1 gives the term a(n) = 10 because (4^5 + 1)/5^2 = 41 is a prime, while for j = 2, (16^17 + 1)/17^2 = 354689 * 2879347902817 is composite. Similarly (256^257 + 1)/257^2 is neither a prime nor the power of a prime, so there is no contribution from the cases j = 2, 3 (see also comments in A127317).
For j = 4 and for any possible other Fermat prime which could be found later, the question is still open, in the sense that it is not actually known if n = 16 * F4 = 1048592 is a term or not. That seems very unlikely, but in order to decide that question for j = 4, one would have to check if (2^1048592 + 1)/65537^2 is a prime or the power of a prime. As this number has 315649 digits, I wonder if it is possible to handle it with the existing primality tests.
(End)

			3 and 4 are not terms because 2^3 + 1 and 2^4 + 1 have only a single prime factor (counted without multiplicity).
6 and 10 are terms because 2^6 + 1 = 5 * 13 and 2^10 + 1 = 5^2 * 41 have two distinct prime factors.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..63
Jack Brennen, Primes of the form (4^p+1)/5^t, Seqfan (Mar 15 2017).
C. Caldwell's The Top Twenty Wagstaff primes.
Mersennewiki, Factorizations Of Cunningham Numbers C+(2,n) (tables).
Samuel S. Wagstaff, The Cunningham Project.
Eric Weisstein's World of Mathematics, Catalan's Conjecture.
Eric Weisstein's World of Mathematics, Zsigmondy Theorem.

Cf. A073936, A107036, A000978, A057182, A127317, A283657, A283364, A092559, A046799.

f[n_] := First[ Transpose[ FactorInteger[2^n + 1]]]; Select[ Range[100], Length[f[ # ]] == 2 & ]
Select[Range[1300],PrimeNu[2^#+1]==2&] (* Harvey P. Dale, Nov 28 2014 *)

isok(k) = #factor(2^k+1)~ == 2; \\ Michel Marcus, Nov 14 2017

A001221(2^a(n) + 1) = 2.

Edited by Robert G. Wilson v, Jan 03 2002

a(40)-a(52) by Giuseppe Coppoletta, May 02 2017

A123628 Smallest prime of the form (q^p+1)/(q+1), where p = prime(n) and q is also prime (q = A123627(n)); or 1 if such a prime does not exist.

Original entry on oeis.org

1, 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 402488219476647465854701, 715827883, 10300379826060720504760427912621791994517454717, 254760179343040585394724919772965278539769280548173566545431025735121201

Offset: 1

Alexander Adamchuk, Oct 03 2006

nonn

a(1) = 1 because such a prime does not exist; (n^2+1) mod (n+1) = 2 for n > 1. a(n) = (A103795(n)^prime(n)+1)/(A103795(n)+1) when A103795(n) is prime. Corresponding smallest primes q such that (q^p+1)/(q+1) is prime, where p = prime(n), are listed in A123627(n) = {0, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 19, 61, 2, 7, 839, 1459, 2, 5, 409, 571, 2, ...}. All Wagstaff primes or primes of form (2^p + 1)/3 belong to a(n). Wagstaff primes are listed in A000979(n) = {3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, ...}. Corresponding indices n such that a(n) = (2^prime(n) + 1)/3 are PrimePi(A000978(n)) = {2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, 31, 39, 43, 46, 65, 69, 126, 267, 380, 495, 762, 1285, 1304, 1364, 1479, 1697, 4469, 8135, 9193, 11065, 11902, 12923, 13103, 23396, 23642, 31850, ...}. All primes with prime indices in the Jacobsthal sequence A001045(n) belong to a(n).

Cf. A123627, A103795, A123487, A123488, A000978, A000979, A001045, A049883, A107036.

a(n) = (A123627(n)^prime(n) + 1) / (A123627(n) + 1).

cp's OEIS Frontend

A049883 Primes in the Jacobsthal sequence (A001045).

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A123627 Smallest prime q such that (q^p+1)/(q+1) is prime, where p = prime(n); or 0 if no such prime q exists.

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A127962 Binary expansion of A000979(n).

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A229524 Numbers k such that (38^k + 1)/39 is prime.

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A125956 Numbers k such that (2^k + 9^k)/11 is prime.

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A073936 Numbers k such that 2^k + 1 is the product of two distinct primes.

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A229663 Numbers n such that (40^n + 1)/41 is prime.

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A230036 Numbers n such that (39^n + 1)/40 is prime.

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