A066651 Primes of the form 2*s + 1, where s is a squarefree number (A005117).

3, 5, 7, 11, 13, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71, 79, 83, 103, 107, 131, 139, 149, 157, 167, 173, 179, 191, 211, 223, 227, 229, 239, 263, 269, 277, 283, 293, 311, 317, 331, 347, 349, 359, 367, 373, 383, 389, 419, 421, 431

Offset: 1

Author

Reinhard Zumkeller, Jan 10 2002

Keywords

• nonn

Comments

For these odd primes delta(p) = A055034(n) = (p-1)/2 is squarefree, and therefore the (Abelian) multiplicative group Modd p (see a comment on A203571 for Modd n, not to be confused with mod n) is guaranteed to be cyclic. This is because the number of Abelian groups of order n (A000688) is 1 precisely for the squarefree numbers A005117. See also A210845. One can in fact prove that the multiplicative group Modd p is cyclic for all primes (the case p=2 is trivial). - Wolfdieter Lang, Sep 24 2012

Examples

			a(13) = A000040(18) = 61 = 2*30+1 = 2*A005117(19)+1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005385, A066652, A066653.

Programs

Mathematica

Select[2 * Select[Range[200], SquareFreeQ] + 1, PrimeQ] (* Amiram Eldar, Feb 22 2021 *)

PARI

isok(p) = isprime(p) && (p>2) && issquarefree((p-1)/2); \\ Michel Marcus, Feb 22 2021