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

1,1

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from _Reinhard Zumkeller)

C. K. Caldwell, Cunningham Chains

Formula

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

Example

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).

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 *)

Prog

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

Crossrefs

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

Sequence in context: A098095 A249953 A180530 * A104213 A178550 A325934

Adjacent sequences: A059497 A059498 A059499 * A059501 A059502 A059503

Keyword

nonn

Author

Labos Elemer, Feb 05 2001

Status

approved