A053703 Primes q of form q=p^w+2 where p is odd prime, w>=2.

11, 29, 83, 127, 6563, 24391, 59051, 161053, 357913, 571789, 1442899, 4782971, 5177719, 14348909, 18191449, 30080233, 73560061, 80062993, 118370773, 127263529, 131872231, 318611989, 344472103

Offset

1,1

Comments

For even w, p=3 is the only prime for which p^w+2 can be prime because all primes greater than 3 have the form 6k+-1. For odd w, only primes p=3 and p=6k-1 need to be considered because all primes of the form p=6k+1 will produce a number p^w+2 that is divisible by 3. - T. D. Noe, Feb 25 2011

Links

T. D. Noe, Table of n, a(n) for n = 1..1005

Formula

Primes of A025475(n)+2 form, excluding 1+2.

a(n) = A053702(n)+2. [R. J. Mathar, Apr 18, 2010]

Example

11=3^2+2, 127=5^3+2, 83=3^4+2, 161051=11^5+2,.. 318611989=683^2+2, 344472103=701^3+2

Mathematica

lst={}; Do[p=Prime[n]; fi=FactorInteger[p-2]; If[Length[fi]==1 && Last[Last[fi]]>1, AppendTo[lst, p]], {n, 20000000}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 25 2011 *)
nn=10^9; t=Table[Select[Table[2 + Prime[i]^k, {i, PrimePi[nn^(1/k)]}], PrimeQ], {k, 2, Log[3, nn]}]; Union[Flatten[t]] (* T. D. Noe, Feb 25 2011 *)

Crossrefs

Cf. A025475.

Sequence in context: A275475 A135064 A179502 * A216559 A099911 A118638

Adjacent sequences: A053700 A053701 A053702 * A053704 A053705 A053706

Keyword

nonn

Author

Labos Elemer, Feb 14 2000

Extensions

Constraint on w added to definition. a(11) appended by R. J. Mathar, Apr 18 2010

Status

approved