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A237188 Smallest member of Sophie Germain pair, where each member of the prime pair is the smallest of its prime triple (p, p+2, p+8).

%I A237188 #27 Aug 07 2025 14:48:15
%S A237188 5,29,2549,6269,41609,259379,418349,492059,514049,521879,819029,
%T A237188 1171199,1659809,1994339,2014139,2325509,2327399,2392139,2420699,
%U A237188 2481179,2844269,3142829,3393359,3637169,3990029
%N A237188 Smallest member of Sophie Germain pair, where each member of the prime pair is the smallest of its prime triple (p, p+2, p+8).
%C A237188 It is not known if there are infinitely many Sophie Germain pairs with this property.
%C A237188 The sequence is infinite under Dickson's conjecture. Aside from a(1) = 5, all terms are 29 or 179 mod 210. - _Charles R Greathouse IV_, Feb 05 2014
%H A237188 Abhiram R Devesh and Charles R Greathouse IV, <a href="/A237188/b237188.txt">Table of n, a(n) for n = 1..10000</a> (first 135 terms from Devesh)
%H A237188 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SophieGermainPrime.html">Sophie Germain Prime</a>
%H A237188 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sophie_Germain_prime">Sophie Germain prime</a>
%e A237188 a(1): p = 5; (2*p)+1 = 11
%e A237188 Prime triples (5,7,13);(11,13,19)
%e A237188 a(2): p = 29; (2*p)+1=59
%e A237188 Prime triples (29,31,37);(59,61,67)
%t A237188 sgpQ[n_]:=Module[{sg=2n+1},AllTrue[Flatten[{sg+{0,2,8},n+{2,8}}], PrimeQ]]; Select[Prime[ Range[ 300000]],sgpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Feb 02 2016 *)
%o A237188 (Python)
%o A237188 from sympy import isprime
%o A237188 print([5] + [n for m in range(29, 10**8, 210) for n in (m, m+150) if isprime(n) and isprime(n+2) and isprime(n+8) and isprime(2*n+1) and isprime(2*n+3) and isprime(2*n+9)]) # _David Radcliffe_, Aug 07 2025
%o A237188 (PARI) is(n)=isprime(n) && isprime(n+2) && isprime(n+8) && isprime(2*n+1) && isprime(2*n+3) && isprime(2*n+9) \\ _Charles R Greathouse IV_, Feb 05 2014
%Y A237188 Cf. A005384.
%K A237188 nonn
%O A237188 1,1
%A A237188 _Abhiram R Devesh_, Feb 04 2014