login
First of four consecutive primes p,q,r,s such that (2*p+q)/5 and (r+2*s)/5 are prime.
3

%I #14 Nov 10 2022 07:43:17

%S 11,1151,33071,33637,55331,57637,75997,90821,97007,100151,112237,

%T 118219,123581,141629,154459,160553,165961,199247,212777,222823,

%U 288361,289511,293677,319993,329471,331697,336101,361799,364537,375371,381467,437279,437693,442571,444461,457837,475751,490877,540781

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

%C 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 Robert Israel, <a href="/A358149/b358149.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 Res:= NULL: count:= 0:

%p q:= 2: r:= 3: s:= 5:

%p while count < 50 do

%p p:= q; q:= r; r:= s; s:= nextprime(s);

%p t:= (2*p+q)/5; u:= (r+2*s)/5;

%p if (t::integer and u::integer and isprime(t) and isprime(u))

%p then

%p count:= count+1; Res:= Res,p;

%p fi

%p od:

%p Res;

%t Select[Partition[Prime[Range[45000]], 4, 1], PrimeQ[(2*#[[1]] + #[[2]])/5] && PrimeQ[(#[[3]] + 2*#[[4]])/5] &][[;; , 1]] (* _Amiram Eldar_, Nov 01 2022 *)

%Y Cf. A358155.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Nov 01 2022