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

Author

Zak Seidov, Feb 27 2004

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

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

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

Programs

Mathematica

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

PARI

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

Extensions

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