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Least number k such that k*p(n)*(k*p(n)+1)-1, k*p(n)*(k*p(n)+1)+1, k*p(n)*(k*p(n)+3)-1 and k*p(n)*(k*p(n)+3)+1 are all primes, two pairs of twin primes, with p(i) = i-th prime.
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%I #6 Jun 05 2021 17:08:36

%S 312,1,3,195,45,48,4884,732,3525,570,1230,2244,930,15555,660,6513,

%T 4656,228,2847,180,2613,21,18,1176,2832,63,3168,4500,12,4740,225,465,

%U 4602,5940,25575,38799,6939,4821,2067,81,21090,9570,18480,558,15762,34836

%N Least number k such that k*p(n)*(k*p(n)+1)-1, k*p(n)*(k*p(n)+1)+1, k*p(n)*(k*p(n)+3)-1 and k*p(n)*(k*p(n)+3)+1 are all primes, two pairs of twin primes, with p(i) = i-th prime.

%e 1*3*(1*3+1)-1=11, 11 and 13 twin primes

%e 1*3*(1*3+3)-1=17, 17 and 19 twin primes so k(2)=1 as p(2)=3

%t lnk[p_]:=Module[{k=1},While[Total[Boole[PrimeQ[{k*p(k*p+1)-1,k*p(k*p+1)+1,k*p(k*p+3)-1,k*p(k*p+3)+1}]]]!=4,k++];k]; Table[lnk[p],{p,Prime[ Range[ 50]]}] (* _Harvey P. Dale_, Jun 05 2021 *)

%K nonn

%O 1,1

%A _Pierre CAMI_, Apr 28 2008