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

1,1

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A000079, A231201, A231557, A264866.

Sequence in context: A242255 A006512 A074304 * A293712 A297674 A072677

Adjacent sequences: A264862 A264863 A264864 * A264866 A264867 A264868

Keyword

nonn

Author

Zhi-Wei Sun, Nov 26 2015

Status

approved