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A358155 First of four consecutive primes p,q,r,s such that (2*p+q)/5, (q+r)/10 and (r+2*s)/5 are prime.

%I A358155 #12 Nov 10 2022 07:43:13
%S A358155 11,2696717,3500381,3989903,4515113,8164073,12451013,18793013,
%T A358155 23567267,24057713,30312409,36391853,44569853,45657881,53442343,
%U A358155 54721253,54944761,56652203,63993803,76763081,90662303,92889127,94670143,105790973,106339481,108988223,117213871,118802533,130741007,145543523
%N A358155 First of four consecutive primes p,q,r,s such that (2*p+q)/5, (q+r)/10 and (r+2*s)/5 are prime.
%C A358155 11 is the only term that is in A007530, because if p is in A007530 (so q = p+2, r = p+6 and s = p+8), one of p, q, r, s, 2*p+q, q+r and r+2*s is divisible by 7.
%H A358155 Robert Israel, <a href="/A358155/b358155.txt">Table of n, a(n) for n = 1..250</a>
%e A358155 a(3) = 3500381 is a term because 3500381, 3500383, 3500407, 3500429 are four consecutive primes with (2*3500381 + 3500383)/5 = 2100229, (3500383 + 3500407)/10 = 700079, and (3500407 + 2*3500429)/5 = 2100253 all prime.
%p A358155 Res:= NULL: count:= 0:
%p A358155 q:= 2: r:= 3: s:= 5:
%p A358155 while count < 40 do
%p A358155   p:= q; q:= r; r:= s; s:= nextprime(s);
%p A358155   t:= (2*p+q)/5; u:= (q+r)/10; v:= (r+2*s)/5;
%p A358155   if (t::integer and u::integer and v::integer and isprime(t) and isprime(u) and isprime(v)) then
%p A358155     count:= count+1; Res:= Res, p;
%p A358155   fi
%p A358155 od:
%p A358155 Res;
%t A358155 Select[Partition[Prime[Range[8.3*10^6]], 4, 1], PrimeQ[(2*#[[1]] + #[[2]])/5] && PrimeQ[(#[[2]] + #[[3]])/10] && PrimeQ[(#[[3]] + 2*#[[4]])/5] &][[;; , 1]] (* _Amiram Eldar_, Nov 01 2022 *)
%Y A358155 Cf. A007530, A358149.
%K A358155 nonn
%O A358155 1,1
%A A358155 _J. M. Bergot_ and _Robert Israel_, Nov 01 2022