A059500 Primes p such that both q=(p-1)/2 and 2p + 1 = 4q + 3 are composite numbers. Intersection of A059456 and A053176.

13, 17, 19, 31, 37, 43, 61, 67, 71, 73, 79, 97, 101, 103, 109, 127, 137, 139, 149, 151, 157, 163, 181, 193, 197, 199, 211, 223, 229, 241, 257, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397, 401, 409, 421, 433, 439, 449

Offset: 1

Author

Labos Elemer, Feb 05 2001

Keywords

• nonn

Comments

Primes which are neither safe nor of Sophie Germain type.

Primes not in Cunningham chains of the first kind. - Alonso del Arte, Jun 30 2005

A010051(a(n))*(1-A156660(a(n)))*(1-A156659(a(n))) = 1; A156878 gives numbers of these numbers <= n. - Reinhard Zumkeller, Feb 18 2009

Examples

			Prime p=17 is here because both 35 and 8 are composite numbers. Such primes fall "out of" any Cunningham chain of first kind (or generate Cunningham chains of 0-length).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from _Reinhard Zumkeller)
• C. K. Caldwell, Cunningham Chains

Crossrefs

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330, A156658.

Programs

Mathematica

Complement[Prime[Range[100]], Select[Prime[Range[100]], PrimeQ[2# + 1] &], Select[Prime[Range[100]], PrimeQ[(# - 1)/2] &]] (Delarte)
Select[Prime[Range[100]],!PrimeQ[q=2#+1]&&!PrimeQ[(#-1)/2]&] (* Zak Seidov, Mar 09 2013 *)

PARI

is(n)=isprime(n)&&!isprime(n\2)&&!isprime(2*n+1) \\ Charles R Greathouse IV, Jan 16 2013

Formula

a(n) ~ n log n. - Charles R Greathouse IV, Jan 16 2013