%I #24 Aug 03 2022 23:22:50

%S 11,29,1811,2531,3373,4153,5927,7121,7127,8419,11743,14347,14419,

%T 17659,26357,26729,33529,43051,57809,61223,81689,87991,99527,99529,

%U 113123,125107,141959,146359,152993,154849,162629,165709,168323,197927,198437,205483,207679,207821,216851,216991,221729,228457

%N First of four consecutive primes p,q,r,s such that the sum of numerator and denominator of p/q + q/r, p/q + r/s, and q/r + r/s, are all prime.

%C First of four consecutive primes p,q,r,s such that p*r + q^2 + q*r, q*s + r^2 + r*s, and p*s + q*r + q*s are all primes.

%H Robert Israel, <a href="/A355696/b355696.txt">Table of n, a(n) for n = 1..10000</a>

%e a(2) = 11 is a term because 11, 13, 17, 19 are consecutive primes: 11/13 + 13/17 = 356/221 with 356 + 221 = 577 prime, 11/13 + 17/19 = 430/247 with 430 + 247 = 677 prime, and 13/17 + 17/19 = 536/323 with 536 + 323 = 859 prime.

%p R:= 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 if isprime(p*r+q^2+q*r) and isprime(q*s+r^2+r*s) and isprime(p*s+q*r+q*s) then

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

%p fi

%p od:

%p R;

%t f[v_] := Module[{p, q, r, s}, {p, q, r, s} = v; PrimeQ[p*r+q^2+q*r] && PrimeQ[q*s+r^2+r*s] && PrimeQ[p*s+q*r+q*s]]; Select[Partition[Prime[Range[20000]], 4, 1], f[#] &][[;;,1]] (* _Amiram Eldar_, Jul 20 2022 *)

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Jul 19 2022