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A065381 Primes not of the form p + 2^k, p prime and k >= 0. 12

%I

%S 2,127,149,251,331,337,373,509,599,701,757,809,877,907,977,997,1019,

%T 1087,1259,1549,1597,1619,1657,1759,1777,1783,1867,1973,2203,2213,

%U 2293,2377,2503,2579,2683,2789,2843,2879,2909,2999,3119,3163,3181,3187,3299

%N Primes not of the form p + 2^k, p prime and k >= 0.

%C Sequence is infinite. For example, Pollack shows that numbers which are 1260327937 mod 2863311360 are not of the form p + 2^k for any prime p and k >= 0, and there are infinitely many primes in this congruence class by Dirichlet's theorem. - _Charles R Greathouse IV_, Jul 20 2014

%H T. D. Noe, <a href="/A065381/b065381.txt">Table of n, a(n) for n = 1..1000</a>

%H P. Pollack, <a href="http://www.math.dartmouth.edu/~ppollack/work.html">Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory</a>, p. 193, ex. 5.1.6, p. 216ff. [?Broken link]

%H P. Pollack, <a href="http://alpha01.dm.unito.it/personalpages/cerruti/ac/notes.pdf">Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory</a>, p. 193, ex. 5.1.6, p. 216ff.

%H Lei Zhou, <a href="http://www.bme.emory.edu/~lzhou/prime/">Between 2^n and primes</a>.

%F A078687(A049084(a(n))) = 0; subsequence of A118958. - _Reinhard Zumkeller_, May 07 2006

%e 127 is a prime, 127-2^0 through 127-2^6 are all nonprimes.

%t fQ[n_] := Block[{k = Floor[Log[2, n]], p = n}, While[k > -1 && ! PrimeQ[p - 2^k], k--]; If[k > 0, True, False]]; Drop[Select[Prime[Range[536]], ! fQ[#] &], {2}] (* _Robert G. Wilson v_, Feb 10 2005; corrected by _Arkadiusz Wesolowski_, May 05 2012 *)

%o (Haskell)

%o a065381 n = a065381_list !! (n-1)

%o a065381_list = filter f a000040_list where

%o f p = all ((== 0) . a010051 . (p -)) $ takeWhile (<= p) a000079_list

%o -- _Reinhard Zumkeller_, Nov 24 2011

%o (PARI) is(p)=my(k=1);while(k<p&&!isprime(p-k),k*=2);if(k>p,return(isprime(p)));0 \\ _Charles R Greathouse IV_, Jul 20 2014

%Y Equals A000040 minus A065380.

%Y Cf. A010051, A006285, A102630, A094076, A156695.

%Y Cf. A098237.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Nov 03 2001

%E Link and cross-reference fixed by _Charles R Greathouse IV_, Nov 09 2008

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Last modified May 26 02:35 EDT 2019. Contains 323579 sequences. (Running on oeis4.)