A059453 Sophie Germain primes (A005384) that are not safe primes (A005385).

2, 3, 29, 41, 53, 89, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901

Offset

1,1

Comments

Except for 2 and 3 these primes are congruent to 5 or 11 modulo 12.

Introducing terms of Cunningham chains of first kind.

Links

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

Chris K. Caldwell, Cunningham Chains.

Formula

A156660(a(n))*(1-A156659(a(n))) = 1. - Reinhard Zumkeller, Feb 18 2009

Example

89 is a term because (89-1)/2 = 44 is not prime, but 2*89 + 1 = 179 is prime.

Mathematica

lst={}; Do[p=Prime[n]; If[ !PrimeQ[(p-1)/2], If[PrimeQ[2*p+1], AppendTo[lst, p]]], {n, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 24 2009 *)
Select[Prime[Range[300]], PrimeQ[2#+1]&&!PrimeQ[(#-1)/2]&] (* Harvey P. Dale, Nov 10 2017 *)

Prog

(Python)
from itertools import count, islice
from sympy import isprime, prime
def A059453_gen(): # generator of terms
    return filter(lambda p:not isprime(p>>1) and isprime(p<<1|1), (prime(i) for i in count(1)))
A059453_list = list(islice(A059453_gen(), 10)) # Chai Wah Wu, Jul 12 2022
(PARI) is(p) = isprime(p) && isprime(2*p+1) && if(p > 2, !isprime((p-1)/2), 1); \\ Amiram Eldar, Jul 15 2024

Crossrefs

Intersection of A005384 and A059456.

Cf. A005385, A053176, A059452, A059453, A059454, A059455, A007700, A005602, A023272, A023302, A023330.

Sequence in context: A284649 A141192 A215135 * A235481 A214889 A137472

Adjacent sequences: A059450 A059451 A059452 * A059454 A059455 A059456

Keyword

nonn

Author

Labos Elemer, Feb 02 2001

Status

approved