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A332848 Primes p such that (3*p+q)/2, (p+3*q)/2, (3*q+r)/2 and (q+3*r)/2 are all prime, where q and r are the next primes after p.

%I A332848 #12 Mar 19 2023 12:16:47
%S A332848 809,15331,51071,59183,59447,95747,125737,224069,442733,471677,521869,
%T A332848 579757,651517,658873,659453,696989,890887,893449,1035707,1114193,
%U A332848 1236517,1271807,1299041,1337593,1435201,1585513,1590383,1672271,1707073,1708363,1817131,1835003,1963309,1992527,2078371,2329597
%N A332848 Primes p such that (3*p+q)/2, (p+3*q)/2, (3*q+r)/2 and (q+3*r)/2 are all prime, where q and r are the next primes after p.
%C A332848 The first of two consecutive primes that are both in A329151.
%C A332848 The first case where a(n+1) is the next prime after a(n), i.e. a(n) and the next two primes are all in A329151, is a(176)=35103361.
%H A332848 Robert Israel, <a href="/A332848/b332848.txt">Table of n, a(n) for n = 1..2000</a>
%e A332848 a(3) = 51071 is in the sequence because p=51071, q=51109, r=51131 are consecutive primes such that (3*p+q)/2=102161, (p+3*q)/2=102199, (3*q+r)/2=102229, (q+3*r)/2=102251 are all prime.
%p A332848 q:= 3: r:= 5:
%p A332848 count:= 0: Res:= NULL:
%p A332848 while count < 100 do
%p A332848   p:= q; q:= r; r:= nextprime(q);
%p A332848   if isprime((3*p+q)/2) and isprime((p+3*q)/2) and isprime((3*q+r)/2)
%p A332848   and isprime((q+3*r)/2) then
%p A332848     count:= count+1; Res:= Res, p;
%p A332848   fi
%p A332848 od:
%p A332848 Res;
%t A332848 Select[Partition[Prime[Range[175000]],3,1],AllTrue[{(3#[[1]]+#[[2]])/2,(#[[1]]+ 3#[[2]])/2,(3#[[2]]+#[[3]])/2,(#[[2]]+3#[[3]])/2},PrimeQ]&][[;;,1]] (* _Harvey P. Dale_, Mar 19 2023 *)
%Y A332848 Cf. A329151.
%K A332848 nonn
%O A332848 1,1
%A A332848 _J. M. Bergot_ and _Robert Israel_, Feb 26 2020