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A297306 Primes p such that q = 4*p+1 and r = (2*p+1)/3 are also primes.

%I A297306 #48 Nov 27 2020 02:08:14
%S A297306 7,43,79,163,673,853,919,1063,1429,1549,1663,2143,2683,3229,3499,4993,
%T A297306 5119,5653,5779,6229,6343,7333,7459,7669,8353,8539,8719,9829,10009,
%U A297306 10243,10303,11383,11689,12583,13399,14149,14653,14923,15649,16603,17053,17389,17749
%N A297306 Primes p such that q = 4*p+1 and r = (2*p+1)/3 are also primes.
%C A297306 This sequence was suggested by _Moshe Shmuel Newman_. It has its source in his study of finite groups.
%H A297306 Alois P. Heinz, <a href="/A297306/b297306.txt">Table of n, a(n) for n = 1..10000</a>
%e A297306 Prime p = 7 is in the sequence because q = 4*7+1 = 29 and r = (2*7+1)/3 = 5 are also primes.
%p A297306 a:= proc(n) option remember; local p; p:= `if`(n=1, 1, a(n-1));
%p A297306       do p:= nextprime(p); if irem(p, 3)=1 and
%p A297306          isprime(4*p+1) and isprime((2*p+1)/3) then break fi
%p A297306       od; p
%p A297306     end:
%p A297306 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jan 07 2018
%t A297306 a[n_] := a[n] = Module[{p}, p = If[n == 1, 1, a[n-1]]; While[True, p = NextPrime[p]; If[Mod[p, 3] == 1 && PrimeQ[4p+1] && PrimeQ[(2p+1)/3], Break[]]]; p];
%t A297306 Array[a, 50] (* _Jean-François Alcover_, Nov 27 2020, after _Alois P. Heinz_ *)
%o A297306 (PARI) isok(p) = isprime(p) && isprime(4*p+1) && iferr(isprime((2*p+1)/3), E, 0); \\ _Michel Marcus_, Nov 27 2020
%Y A297306 Cf. A000040.
%Y A297306 Intersection of A023212 and A104163.
%K A297306 nonn
%O A297306 1,1
%A A297306 _David S. Newman_, Jan 04 2018
%E A297306 More terms from _Alois P. Heinz_, Jan 07 2018