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A358149 First of four consecutive primes p,q,r,s such that (2*p+q)/5 and (r+2*s)/5 are prime.

%I A358149 #14 Nov 10 2022 07:43:17
%S A358149 11,1151,33071,33637,55331,57637,75997,90821,97007,100151,112237,
%T A358149 118219,123581,141629,154459,160553,165961,199247,212777,222823,
%U A358149 288361,289511,293677,319993,329471,331697,336101,361799,364537,375371,381467,437279,437693,442571,444461,457837,475751,490877,540781
%N A358149 First of four consecutive primes p,q,r,s such that (2*p+q)/5 and (r+2*s)/5 are prime.
%C A358149 Dickson's conjecture implies there are infinitely many terms where q = p+2, r = p+6 and s = p+8; the first two of these are 11 and 55331.
%H A358149 Robert Israel, <a href="/A358149/b358149.txt">Table of n, a(n) for n = 1..10000</a>
%e A358149 a(3) = 33071 is a term because 33071, 33073, 33083, 33091 are four consecutive primes with (2*33071+33073)/5 = 19843 and (33083+2*33091)/5 = 19853 prime.
%p A358149 Res:= NULL: count:= 0:
%p A358149 q:= 2: r:= 3: s:= 5:
%p A358149 while count < 50 do
%p A358149   p:= q; q:= r; r:= s; s:= nextprime(s);
%p A358149   t:= (2*p+q)/5; u:= (r+2*s)/5;
%p A358149   if (t::integer and u::integer and isprime(t) and isprime(u))
%p A358149    then
%p A358149     count:= count+1; Res:= Res,p;
%p A358149   fi
%p A358149 od:
%p A358149 Res;
%t A358149 Select[Partition[Prime[Range[45000]], 4, 1], PrimeQ[(2*#[[1]] + #[[2]])/5] && PrimeQ[(#[[3]] + 2*#[[4]])/5] &][[;; , 1]] (* _Amiram Eldar_, Nov 01 2022 *)
%Y A358149 Cf. A358155.
%K A358149 nonn
%O A358149 1,1
%A A358149 _J. M. Bergot_ and _Robert Israel_, Nov 01 2022