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

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

Author

Vladimir Shevelev, Mar 13 2017

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A002587, A046799, A283364, A073936, A000978, A127317, A092559, A019434, A066263.

Contains 4*A057182.

Programs

Maple

# 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

Mathematica

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

PARI

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

Extensions

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