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A264865
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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.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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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.
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A000040, A000079, A231201, A231557, A264866.
Sequence in context: A242255 A006512 A074304 * A293712 A297674 A072677
Adjacent sequences: A264862 A264863 A264864 * A264866 A264867 A264868
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KEYWORD
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nonn
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AUTHOR
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Zhi-Wei Sun, Nov 26 2015
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STATUS
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approved
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