This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p1*p2+6]&&PrimeQ[p1*p2-6],(*Print[p1];*)AppendTo[lst,p1]],{n,8!}];lst
lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p1*p2-4]&&PrimeQ[p1*p2-6],(*Print[p1];*)AppendTo[lst,p1]],{n,8!}];lst
ltp[{a_,b_}]:=b-a==2&&AllTrue[a*b-{4,6},PrimeQ]; Select[Partition[Prime[ Range[ 10000]],2,1],ltp][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 07 2017 *)
a(3) = 11 is a term because the next prime is 13 and 2*11+3*13 = 61 and 3*11+2*13 = 59 are prime.
R:= NULL: count:= 0:
q:= 2:
while count < 100 do
p:= q; q:= nextprime(p);
if isprime(2*p+3*q) and isprime(3*p+2*q) then
count:= count+1; R:= R, p
fi
od:
R;
Select[Partition[Prime[Range[500]],2,1],AllTrue[{2#[[1]]+3#[[2]],3#[[1]]+2#[[2]]},PrimeQ]&][[;;,1]] (* Harvey P. Dale, Jun 11 2025 *)
isok(p) = isprime(p) && (q=nextprime(p+1)) && isprime(2*p+3*q) && isprime(3*p+2*q); \\ Michel Marcus, Feb 07 2021
3 and 5 are twin primes, but 5*3+4 = 19 and 21 are not twin primes. 11 and 13 are twin primes, 5*11+4 = 59 and 61 are twin primes, but 5*59+4 = 299 and 301 are not twin primes. 5 and 7 are twin primes, 5*5+4 = 29 and 31 are twin primes, 5*29+4 = 149 and 151 are twin primes, but 5*149+4 = 749 and 751 are not twin primes. 1720949 and 1720951 are twin primes, 5*1720949+4 = 8604749 and 8604751 are twin primes, 5*8604749+4 = 43023749 and 43023751 are twin primes, 5*8604749+4 = 215118749 and 215118751 are twin primes, 5*1075593749+4 = 1075593749 and 1075593751 are not twin primes.
V:= [3,0,0,0]:
count:= 1:
for p from 5 by 6 while count < 4 do
if isprime(p) and isprime(p+2) then
ct:= 1:
q:= p;
do
q:= 5*q+4;
if not (isprime(q) and isprime(q+2)) then break fi;
ct:= ct+1;
od;
if V[ct] = 0 then V[ct]:= p; count:= count+1; fi;
fi
od:
V;
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