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A264866 Primes of the form 2^x + y (x >= 0 and 0 <= y < 2^x) such that all the numbers 2^(x+a) + (y-a) (0 < a <= y) are composite.

%I A264866 #11 Nov 27 2015 14:07:01
%S A264866 2,3,5,11,13,17,19,23,41,71,131,149,257,277,523,1117,2053,2161,2237,
%T A264866 2251,2999,4099,5237,8233,8243,16453,16553,32771,32779,32783,32789,
%U A264866 32797,32801,32839,32843,32917,33623,65537,65539,65543,65563,65599,65651,72497,131129,131267,134777,262147,262151,264959
%N A264866 Primes of the form 2^x + y (x >= 0 and 0 <= y < 2^x) such that all the numbers 2^(x+a) + (y-a) (0 < a <= y) are composite.
%C A264866 Conjecture: The sequence has infinitely many terms.
%C A264866 This is motivated by part (i) of the conjecture in A231201 and the conjecture in A264865.
%H A264866 Zhi-Wei Sun, <a href="/A264866/b264866.txt">Table of n, a(n) for n = 1..62</a>
%H A264866 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;85566cc1.1311">Write n = k + m with 2^k + m prime</a>, a message to Number Theory List, Nov. 16, 2013.
%H A264866 Z.-W. Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n+ bn modulo m</a>, arXiv:1312.1166 [math.NT], 2013-2014.
%H A264866 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014-2015.
%e A264866 a(4) = 11 since 11 = 2^3 + 3 is a prime with 3 < 2^3, and 2^4 + 2 = 18, 2^5 + 1 = 33 and 2^6 + 0 = 64 are all composite.
%t A264866 p[n_]:=p[n]=Prime[n]
%t A264866 x[n_]:=x[n]=Floor[Log[2,p[n]]]
%t A264866 y[n_]:=y[n]=p[n]-2^(x[n])
%t A264866 n=0;Do[Do[If[PrimeQ[2^(x[k]+a)+y[k]-a],Goto[aa]],{a,1,y[k]}];n=n+1;Print[n," ",p[k]];Label[aa];Continue,{k,1,23226}]
%Y A264866 Cf. A000040, A000079, A231201, A231557, A264865.
%K A264866 nonn
%O A264866 1,1
%A A264866 _Zhi-Wei Sun_, Nov 26 2015