A153209 Primes of the form 2*p+1 where p is prime and p+1 is squarefree.

5, 11, 59, 83, 227, 347, 563, 1019, 1283, 1307, 1523, 2459, 2579, 2819, 2963, 3803, 3947, 4259, 4547, 5387, 5483, 6779, 6827, 7187, 8147, 9587, 10667, 10883, 11003, 12107, 12227, 12539, 12659, 13043, 13163, 14243, 14387, 15683, 16139, 16187

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Dec 20 2008

Keywords

• nonn

Comments

Subsequence of A005385.

Examples

			For p = 2 (the only case with p+1 odd), 2*p+1 = 5 is prime and p+1 = 3 is squarefree, so 5 is in the sequence. For p = 3, 2*p+1 = 7 is prime and p+1 = 4 is not squarefree, so 7 is not in the sequence.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005117 (squarefree numbers), A005385 (safe primes p: (p-1)/2 is also prime), A153207, A153208, A153210.

Programs

Magma

[ q: p in PrimesUpTo(8100) | IsSquarefree(p+1) and IsPrime(q) where q is 2*p+1 ];

Mathematica

lst = {}; Do[p = Prime[n]; If[PrimeQ[Floor[p/2]] && SquareFreeQ[Ceiling[p/2]], AppendTo[lst, p]], {n, 7!}]; lst
Select[2#+1&/@Select[Prime[Range[2000]],SquareFreeQ[#+1]&],PrimeQ] (* Harvey P. Dale, Aug 02 2024 *)

Extensions

Edited by Klaus Brockhaus, Dec 24 2008

Mathematica updated by Jean-François Alcover, Jul 04 2013