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.

5, 7, 13, 19, 31, 47, 61, 71, 101, 211, 239, 241, 271, 281, 311, 331, 379, 421, 449, 491, 617, 619, 631, 751, 797, 827, 853, 863, 883, 971, 991, 1009, 1051, 1117, 1171, 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

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.

See also A264866 for a related conjecture.

Examples

			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.
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• 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, A264866.

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,x[k]}];n=n+1;Print[n," ",p[k]];Label[aa];Continue,{k,1,283}]