

A283657


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


3



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
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OFFSET

1,3


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 MihăilescuCatalan'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).


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.


EXAMPLE

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.


MAPLE

# this uses A002587[i] for i<=500, e.g., from the bfile 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 *)


PROG

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


CROSSREFS

Cf. A002587, A046799, A283364, A073936, A000978, A127317, A092559, A019434, A066263.
Contains 4*A057182.
Sequence in context: A102450 A023782 A114522 * A053432 A261888 A154125
Adjacent sequences: A283654 A283655 A283656 * A283658 A283659 A283660


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, 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


STATUS

approved



