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A118955 Numbers of the form 2^k + prime. 18


%S 3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20,21,23,24,25,27,29,30,31,

%T 32,33,34,35,37,38,39,41,42,43,44,45,47,48,49,51,53,54,55,57,59,60,61,

%U 62,63,65,66,67,68,69,71,72,73,74,75,77,79,80,81,83,84,85,87,89,90,91,93

%N Numbers of the form 2^k + prime.

%C A109925(a(n)) > 0, complement of A118954;

%C The lower density is at least 0.09368 (Pintz) and upper density is at most 0.49095 (Habsieger & Roblot). The density, if it exists, is called Romanov's constant. Romani conjectures that it is around 0.434. - _Charles R Greathouse IV_, Mar 12 2008

%H Charles R Greathouse IV, <a href="/A118955/b118955.txt">Table of n, a(n) for n = 1..10000</a>.

%H Laurent Habsieger and Xavier-Francois Roblot, <a href="http://journals.impan.gov.pl/aa/Inf/122-1-4.html">On integers of the form p + 2^k</a>, Acta Arithmetica 122:1 (2006), pp. 45-50.

%H J. Pintz, <a href="http://dx.doi.org/10.1007/s10474-006-0060-6">A note on Romanov's constant</a>, Acta Mathematica Hungarica 112:1-2 (2006), pp. 1-14.

%H F. Romani, <a href="http://dx.doi.org/10.1007/BF02576468">Computations concerning primes and powers of two</a>, Calcolo 20 (1983), pp. 319-336.

%t Select[Range[100], (For[r=False; k=1, #>k, k*=2, If[PrimeQ[#-k], r=True]]; r)& ] (* _Jean-Fran├žois Alcover_, Dec 26 2013, after _Charles R Greathouse IV_ *)

%o (PARI) is(n)=my(k=1);while(n>k,if(isprime(n-k),return(1),k*=2));0 \\ _Charles R Greathouse IV_, Mar 12 2008

%o (Haskell)

%o a118955 n = a118955_list !! (n-1)

%o a118955_list = filter f [1..] where

%o f x = any (== 1) $ map (a010051 . (x -)) $ takeWhile (< x) a000079_list

%o -- _Reinhard Zumkeller_, Jan 03 2014

%Y Subsequence of A081311; A118957 is a subsequence.

%Y Cf. A156695, A010051, A000079.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, May 07 2006

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Last modified November 27 15:11 EST 2020. Contains 338683 sequences. (Running on oeis4.)