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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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, 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
Sequence in context: A220815 A171600 A126148 * A038933 A042998 A091317
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 26 2015
STATUS
approved

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Last modified June 24 21:56 EDT 2024. Contains 373690 sequences. (Running on oeis4.)