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.
a(3) = 2 since 2, 2*2+3=7 and 2*7+3=17 are primes.
a(3) = 5 since 5, f(5) = 29 and f(29) = 149 are primes when f(x) = 5x+4.
t[p_] := Block[{c=1, q = 5*p+4}, While[ PrimeQ@q, q = 5*q + 4; c++]; c]; a[n_] := Block[{p = 2}, While[t[p] < n, p = NextPrime@ p]; p]; Array[a, 8] (* Giovanni Resta, Mar 21 2017 *)
a(3)=13 since 13, f(13)=67 and f(67)=337 are primes when f(x) = 5x+2.
c[p_] := Block[{k = 1, q = 5*p+2}, While[ PrimeQ[q], q = 5*q+2; k++]; k]; a[n_] := Block[{p = 2}, While[c[p] < n, p = NextPrime@ p]; p]; Array[a, 7] (* Giovanni Resta, Mar 21 2017 *)
a(3) = 3 since 3, f(3) = 13 and f(13) = 43 are primes when f(x) = 3*x + 4.
c[p_] := Block[{k=1, q=3*p + 4}, While[PrimeQ[q], q=3*q+4; k++]; k]; a[n_] := Block[{p = 2}, While[c[p] < n, p = NextPrime[p]]; p]; Array[a, 7] (* Giovanni Resta, Mar 22 2017 *)
a(3) = 2 since 2, f(2) = 11, and f(11) = 47 are primes when f(x) = 4*x + 3.
c[p_] := Block[{k=1, q=4*p+3}, While[ PrimeQ[q], q=4*q+3; k++]; k]; a[n_] := Block[ {p=2}, While[c[p] < n, p = NextPrime@ p]; p]; Array[a, 9] (* Giovanni Resta, Mar 21 2017 *)
has(p,n)=for(i=2,n, if(!isprime(p=4*p+3), return(0))); 1 a(n)=forprime(p=2,, if(has(p,n), return(p))) \\ Charles R Greathouse IV, Jan 20 2017
a(3) = 13 since 7, f(7) = 41, and f(41) = 211 are primes when f(x) = 5*x + 6.
c[p_] := Block[{k=1, q = 5*p+6}, While[PrimeQ[q], q = 5*q+6; k++]; k]; a[n_] := Block[{p = 2}, While[c[p] < n, p = NextPrime[p]]; p]; Array[a, 7] (* Giovanni Resta, Mar 22 2017 *)
a(3) = 2 since 2, f(2) = 17, and f(17) = 107 are primes when f(x) = 6*x + 5.
c[p_] := Block[{k=1, q=6*p+5}, While[ PrimeQ[q], q = 6*q+5; k++]; k]; a[n_] := Block[ {p=2}, While[c[p] < n, p = NextPrime[p]]; p]; Array[a, 7] (* Giovanni Resta, Mar 22 2017 *)
a(2) = 3 because 3 is greatest prime factor of 2^2+2. a(3)=11 because 3^2+2 is prime.
a[1]=2; a[n_] := a[n]=FactorInteger[a[n-1]^2+2][[ -1, 1]] NestList[FactorInteger[#^2+2][[-1,1]]&,2,15] (* Harvey P. Dale, Jun 21 2022 *)
a(4) = 5 because 5 is the beginning of 4 primes in succession: 5, 3*5 - 2 = 13 is prime, 3*13 - 2 = 37 is prime, 3*37 - 2 = 109 is prime.
c[p_] := Block[{k=1, q = 3 p - 2}, While[PrimeQ[q], q = 3 q - 2; k++]; k]; a[n_] := Block[{p=2}, While[c[p] < n, p = NextPrime[p]]; p]; Array[a, 7]
a(n)={x=1;k=1;while(k==1,m=0;y=x;while(isprime(y),m++;if(m==n,k=x);y=3*y-2);x++);k;}
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