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.
ppp[{p_,a_}]:=Module[{n=a+2},While[!PrimeQ[p*n+1],n=n+2];{p*n,n}]; NestList[ ppp,{2,2},60][[All,2]] (* Harvey P. Dale, Aug 12 2017 *)
a[1] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 2}, While[ !PrimeQ[k] || ! PrimeQ[ k * Times @@ Table[ a[i], {i, 1, n - 1}] + 2], k += 2]; k]; Table[ a[n], {n, 1, 49}]
nxt[{t_,a_}]:=Module[{p=NextPrime[a]},While[!PrimeQ[t*p+2],p= NextPrime[ p]];{t*p,p}]; NestList[nxt,{1,1},50][[All,2]] (* Harvey P. Dale, Jan 09 2022 *)
For 1, 3, 5, 7: 1+2 = 3, 1*3+2 = 5, 1*3*5+2 = 17, 1*3*5*7+2 = 107 are primes. - _Daniel Forgues_, Dec 20 2012
f[s_List] := Block[{k = 1, p = Times @@ s}, While[ MemberQ[s, k] || !PrimeQ[k*p + 2], k += 2]; Append[s, k]]; Nest[f, {1}, 62] (* Robert G. Wilson v, Dec 24 2012 *)