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

1,1

Comments

Conjecture: The sequence has infinitely many terms.

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

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A220815 A171600 A126148 * A038933 A042998 A091317

Adjacent sequences: A264863 A264864 A264865 * A264867 A264868 A264869

Keyword

nonn

Author

Zhi-Wei Sun, Nov 26 2015

Status

approved