A066263 -id:A066263 - cp's OEIS frontend

A092559 Numbers k such that 2^k + 1 is a semiprime.

3, 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

Zak Seidov, Feb 27 2004

nonn

Thanks to the recently found factor of F_14 (see A093179), we know that 16384 is not in the sequence. First unknown: 16768. - Don Reble, Mar 28 2010
The big prime factors for "5807" and all smaller entries have been proved prime; the rest (as far as I know) are probable primes. - Don Reble, Mar 28 2010
From Giuseppe Coppoletta, May 09 2017: (Start)
As 3 divides 2^a(n) + 1 for any odd a(n), all odd terms are prime and they are exactly the Wagstaff numbers (A000978) or also the prime Jacobsthal indices (A107036).
All terms from a(51) onwards refer to probabilistic primality tests for 2^a(n) + 1 (see Caldwell's link for the list of the largest certified Wagstaff primes).
For the close relationship between this sequence and the Fermat numbers, see comments in A073936. The only difference is that here a term can be the square of a prime p, and by the Mihăilescu Theorem (also known as Catalan's conjecture, see link) that implies p = a(n) = 3. So, excluding a(1) = 3, they must coincide.
As for A073936, after a(57), the values 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 are possibly much further along in the numbering (see comments in A000978).
(End).
The powers of 2 in this sequence (that correspond to semiprime Fermat numbers) are k = 2^m with m = 5, 6, 7, 8, and no more below 20. - Amiram Eldar, Jun 18 2022

			11 is a term because 2^11 + 1 = 3 * 683.
3 is a term because 2^3 + 1 = 3^2.
10 is not a term because 2^10 + 1 = 5^2 * 41.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..57
C. Caldwell's The Top Twenty Wagstaff primes.
S. S. Wagstaff, Jr., The Cunningham Project.
Eric W. Weisstein, MathWorld: Catalan's Conjecture.

Cf. A085724, A092558, A092561, A092562, A000978, A107036, A066263.

Cf. A073936. - R. J. Mathar, Sep 08 2008

Select[Range@ 200, PrimeOmega[2^# + 1] == 2 &] (* Michael De Vlieger, May 09 2017 *)

isok(n) = bigomega(2^n+1) == 2; \\ Michel Marcus, Oct 05 2013

More terms from Cunningham project, Mar 23 2004
More terms from Don Reble, Mar 28 2010
a(49)-a(52) from Giuseppe Coppoletta, May 08 2017

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

A283657 Numbers m such that 2^m + 1 has at most 2 distinct prime factors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 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

Offset: 1

Vladimir Shevelev, Mar 13 2017

nonn

Using comment in A283364, note that if a(n) is odd > 9, then it is prime.
503 <= a(41) <= 596. - Robert Israel, Mar 13 2017
Could (4^p + 1)/5^t be prime, where p is prime, 5^t is the highest power of 5 dividing 4^p + 1, other than for p=2, 3 and 5? - Vladimir Shevelev, Mar 14 2017
In his message to seqfans from Mar 15 2017, Jack Brennen beautifully proved that there are no more primes of such form. From his proof one can see also that there are no terms of the form 2*p > 10 in the sequence. - Vladimir Shevelev, Mar 15 2017
Where A046799(n)=2. - Robert G. Wilson v, Mar 15 2017
From Giuseppe Coppoletta, May 16 2017: (Start)
The only terms that are not in A066263 are those m giving 2^m + 1 = prime (i.e. m = 0 and any number m such that 2^m + 1 is a Fermat prime) and the values of m giving 2^m + 1 = power of a prime, giving m = 3 as the only possible case (by Mihăilescu-Catalan's result, see links).
For the relation with Fermat numbers and for other possible terms to check, see comments in A073936 and A066263.
All terms after a(59) 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(65), the values 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 further along in the numbering (see comments in A000978).
(End).

			0 is a term as 2^0 + 1 = 2 is a prime.
10 is a term as 2^10 + 1 = 5^2 * 41.
14 is not a term as 2^14 + 1 = 5 * 29 * 113.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..65
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. A002587, A046799, A283364, A073936, A000978, A127317, A092559, A019434, A066263.

Contains 4*A057182.

# this uses A002587[i] for i<=500, e.g., from the b-file for that sequence
count:= 0:
for i from 0 to 500 do
  m:= 0;
  r:= (2^i+1);
  if i::odd then
    m:= 1;
    r:= r/3^padic:-ordp(r,3);
  elif i > 2 then
    q:= max(numtheory:-factorset(i));
    if q > 2 then
      m:= 1;
      r:= r/B[i/q]^padic:-ordp(r,A002587[i/q]);
    fi
  fi;
  if r mod B[i] = 0 then m:= m+1;
      j:= padic:-ordp(r, A002587[i]);
      r:= r/B[i]^j;
  fi;
  mmax:= m;
  if isprime(r) then m:= m+1; mmax:= m
  elif r > 1 then mmax:= m+2
  fi;
  if mmax <= 2 or (m <= 1 and m + nops(numtheory:-factorset(r)) <= 2) then
       count:= count+1;
     A[count]:= i;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Mar 13 2017

Select[Range[0, 313], PrimeNu[2^# + 1]<3 &] (* Indranil Ghosh, Mar 13 2017 *)

for(n=0, 313, if(omega(2^n + 1)<3, print1(n,", "))) \\ Indranil Ghosh, Mar 13 2017

a(16)-a(38) from Peter J. C. Moses, Mar 13 2017
a(39)-a(40) from Robert Israel, Mar 13 2017
a(41)-a(65) from Giuseppe Coppoletta, May 08 2017

A242076 Numbers k for which (2^k + 1)/F is prime where F is a Fermat number.

Original entry on oeis.org

3, 5, 6, 7, 11, 12, 13, 17, 19, 20, 23, 28, 31, 40, 43, 61, 79, 92, 96, 101, 104, 127, 148, 167, 191, 199, 313, 347, 356, 596, 692, 701, 1004, 1228, 1268, 1709, 2617, 3539, 3824, 5807, 10501, 10691, 11279, 12391, 14479, 42737

Offset: 1

J. Lowell, May 03 2014

Conjecture: 6 is the only term whose prime factorization contains a single 2.
The largest odd divisor of each term is prime, that is, subsequence of A038550. - J. Lowell, Apr 13 2018
This sequence contains only certain terms from A092559 and certain multiples of 32. - Jon E. Schoenfield, Apr 18 2018 [with thanks to J. Lowell]

			12 is a term because (2^12 + 1)/17 = 241, a prime number.

Cf. A000215 (Fermat numbers), A066263.

def a(n):
    num = 2^n + 1
    k = 0
    while k < log(n, 2):
        if num % (2^(2^k) + 1) == 0 and is_prime(Integer(num/(2^(2^k)+1))):
            return True
        k = k + 1
    return False          # Ralf Stephan, May 15 2014

More terms from Ralf Stephan, May 15 2014
a(40)-a(46) from Jon E. Schoenfield, Apr 14 2018
Wrong property removed by J. Lowell, Apr 14 2018

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A092559 Numbers k such that 2^k + 1 is a semiprime.

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

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A283657 Numbers m such that 2^m + 1 has at most 2 distinct prime factors.

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A242076 Numbers k for which (2^k + 1)/F is prime where F is a Fermat number.

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