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A264865 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 <= x) are composite.

%I A264865 #10 Nov 26 2015 23:46:03
%S A264865 5,7,13,19,31,47,61,71,101,211,239,241,271,281,311,331,379,421,449,
%T A264865 491,617,619,631,751,797,827,853,863,883,971,991,1009,1051,1117,1171,
%U A264865 1217,1277,1291,1297,1301,1321,1327,1429,1453,1471,1483,1487,1531,1567,1607,1627,1637,1667,1669,1697,1709,1723,1747,1801,1847
%N A264865 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 <= x) are composite.
%C A264865 Conjecture: The sequence has infinitely many terms.
%C A264865 This is motivated by part (i) of the conjecture in A231201.
%C A264865 See also A264866 for a related conjecture.
%H A264865 Zhi-Wei Sun, <a href="/A264865/b264865.txt">Table of n, a(n) for n = 1..10000</a>
%H A264865 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 A264865 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 A264865 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 A264865 a(1) = 5 since 5 = 2^2 + 1 is a prime with 1 < 2^2, and 2^0 + 3 = 2^1 + 2 = 4 is composite.
%e A264865 a(3) = 13 since 13 = 2^3 + 5 is a prime with 5 < 2^3, and 2^0 + 8 = 2^1 + 7 = 9 and 2^2 + 6 = 10 are both composite.
%t A264865 p[n_]:=p[n]=Prime[n]
%t A264865 x[n_]:=x[n]=Floor[Log[2,p[n]]]
%t A264865 y[n_]:=y[n]=p[n]-2^(x[n])
%t A264865 n=0;Do[Do[If[PrimeQ[2^(x[k]-a)+y[k]+a],Goto[aa]],{a,1,x[k]}];n=n+1;Print[n," ",p[k]];Label[aa];Continue,{k,1,283}]
%Y A264865 Cf. A000040, A000079, A231201, A231557, A264866.
%K A264865 nonn
%O A264865 1,1
%A A264865 _Zhi-Wei Sun_, Nov 26 2015