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

%I A059500 #27 Apr 03 2023 10:36:09
%S A059500 13,17,19,31,37,43,61,67,71,73,79,97,101,103,109,127,137,139,149,151,
%T A059500 157,163,181,193,197,199,211,223,229,241,257,269,271,277,283,307,311,
%U A059500 313,317,331,337,349,353,367,373,379,389,397,401,409,421,433,439,449
%N A059500 Primes p such that both q=(p-1)/2 and 2p + 1 = 4q + 3 are composite numbers. Intersection of A059456 and A053176.
%C A059500 Primes which are neither safe nor of Sophie Germain type.
%C A059500 Primes not in Cunningham chains of the first kind. - _Alonso del Arte_, Jun 30 2005
%C A059500 A010051(a(n))*(1-A156660(a(n)))*(1-A156659(a(n))) = 1; A156878 gives numbers of these numbers <= n. - _Reinhard Zumkeller_, Feb 18 2009
%H A059500 Amiram Eldar, <a href="/A059500/b059500.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from _Reinhard Zumkeller)
%H A059500 C. K. Caldwell, <a href="https://t5k.org/glossary/page.php/CunninghamChain">Cunningham Chains</a>
%F A059500 a(n) ~ n log n. - _Charles R Greathouse IV_, Jan 16 2013
%e A059500 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).
%t A059500 Complement[Prime[Range[100]], Select[Prime[Range[100]], PrimeQ[2# + 1] &], Select[Prime[Range[100]], PrimeQ[(# - 1)/2] &]] (Delarte)
%t A059500 Select[Prime[Range[100]],!PrimeQ[q=2#+1]&&!PrimeQ[(#-1)/2]&] (* _Zak Seidov_, Mar 09 2013 *)
%o A059500 (PARI) is(n)=isprime(n)&&!isprime(n\2)&&!isprime(2*n+1) \\ _Charles R Greathouse IV_, Jan 16 2013
%Y A059500 Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330, A156658.
%K A059500 nonn
%O A059500 1,1
%A A059500 _Labos Elemer_, Feb 05 2001