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.

2, 3, 5, 11, 13, 17, 19, 23, 41, 71, 131, 149, 257, 277, 523, 1117, 2053, 2161, 2237, 2251, 2999, 4099, 5237, 8233, 8243, 16453, 16553, 32771, 32779, 32783, 32789, 32797, 32801, 32839, 32843, 32917, 33623, 65537, 65539, 65543, 65563, 65599, 65651, 72497, 131129, 131267, 134777, 262147, 262151, 264959

Offset: 1

Author

Zhi-Wei Sun, Nov 26 2015

Keywords

• nonn

Comments

Conjecture: The sequence has infinitely many terms.

This is motivated by part (i) of the conjecture in A231201 and the conjecture in A264865.

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..62
• Zhi-Wei Sun, Write n = k + m with 2^k + m prime, a message to Number Theory List, Nov. 16, 2013.
• Z.-W. Sun, On a^n+ bn modulo m, arXiv:1312.1166 [math.NT], 2013-2014.
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2015.

Crossrefs

Cf. A000040, A000079, A231201, A231557, A264865.

Programs

Mathematica

p[n_]:=p[n]=Prime[n]
x[n_]:=x[n]=Floor[Log[2,p[n]]]
y[n_]:=y[n]=p[n]-2^(x[n])
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}]