cp's OEIS Frontend

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

%I A092559 #47 Feb 16 2025 08:32:52
%S A092559 3,5,6,7,11,12,13,17,19,20,23,28,31,32,40,43,61,64,79,92,101,104,127,
%T A092559 128,148,167,191,199,256,313,347,356,596,692,701,1004,1228,1268,1709,
%U A092559 2617,3539,3824,5807,10501,10691,11279,12391,14479,42737,83339,95369,117239
%N A092559 Numbers k such that 2^k + 1 is a semiprime.
%C A092559 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
%C A092559 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
%C A092559 From _Giuseppe Coppoletta_, May 09 2017: (Start)
%C A092559 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).
%C A092559 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).
%C A092559 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.
%C A092559 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).
%C A092559 (End).
%C A092559 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
%H A092559 Giuseppe Coppoletta, <a href="/A092559/b092559.txt">Table of n, a(n) for n = 1..57</a>
%H A092559 C. Caldwell's The Top Twenty <a href="https://t5k.org/top20/page.php?id=67">Wagstaff primes</a>.
%H A092559 S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>.
%H A092559 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/CatalansConjecture.html">MathWorld: Catalan's Conjecture</a>.
%e A092559 11 is a term because 2^11 + 1 = 3 * 683.
%e A092559 3 is a term because 2^3 + 1 = 3^2.
%e A092559 10 is not a term because 2^10 + 1 = 5^2 * 41.
%t A092559 Select[Range@ 200, PrimeOmega[2^# + 1] == 2 &] (* _Michael De Vlieger_, May 09 2017 *)
%o A092559 (PARI) isok(n) = bigomega(2^n+1) == 2; \\ _Michel Marcus_, Oct 05 2013
%Y A092559 Cf. A085724, A092558, A092561, A092562, A000978, A107036, A066263.
%Y A092559 Cf. A073936. - _R. J. Mathar_, Sep 08 2008
%K A092559 nonn
%O A092559 1,1
%A A092559 _Zak Seidov_, Feb 27 2004
%E A092559 More terms from Cunningham project, Mar 23 2004
%E A092559 More terms from _Don Reble_, Mar 28 2010
%E A092559 a(49)-a(52) from _Giuseppe Coppoletta_, May 08 2017