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

2, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 907, 977, 997, 1019, 1087, 1259, 1549, 1597, 1619, 1657, 1759, 1777, 1783, 1867, 1973, 2203, 2213, 2293, 2377, 2503, 2579, 2683, 2789, 2843, 2879, 2909, 2999, 3119, 3163, 3181, 3187, 3299

Offset

1,1

Comments

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

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

P. Pollack, Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory, p. 193, ex. 5.1.6, p. 216ff. [?Broken link]

P. Pollack, Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory, p. 193, ex. 5.1.6, p. 216ff.

Lei Zhou, Between 2^n and primes.

Formula

A078687(A049084(a(n))) = 0; subsequence of A118958. - Reinhard Zumkeller, May 07 2006

Example

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

Mathematica

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 *)

Prog

(Haskell)
a065381 n = a065381_list !! (n-1)
a065381_list = filter f a000040_list where
   f p = all ((== 0) . a010051 . (p -)) $ takeWhile (<= p) a000079_list
-- Reinhard Zumkeller, Nov 24 2011
(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

Crossrefs

Equals A000040 minus A065380.

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

Cf. A098237.

Sequence in context: A364654 A139904 A167414 * A274123 A141928 A343184

Adjacent sequences: A065378 A065379 A065380 * A065382 A065383 A065384

Keyword

nonn

Author

Reinhard Zumkeller, Nov 03 2001

Extensions

Link and cross-reference fixed by Charles R Greathouse IV, Nov 09 2008

Status

approved