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.
k = 1; a = {}; Do[If[PrimeQ[k n + 1], k = k n; AppendTo[a, k + 1]], {n, 1, 1000}]; a (* Artur Jasinski, Sep 19 2008 *)
a(1)=1 because a(0) is not defined and 2*1 + 1 = 3 is prime; a(2)=2 because 2*1*2 + 1 = 5 is prime; a(3)=3 because 2*1*2*3 + 1 = 13 is prime; a(4) is not 4 because 2*1*2*3*4 + 1 = 49 is not prime, but a(4)=5 works because 2*1*2*3*5 + 1 = 61 is prime.
k = 2; a = {}; Do[If[PrimeQ[k n + 1], k = k n; AppendTo[a, n]], {n, 1, 3000}]; a (* Artur Jasinski *)
nxt[{p_,a_}]:=Module[{k=a+1},While[!PrimeQ[p*k+1],k++];{p*k,k}]; NestList[ nxt,{2,1},60][[All,2]] (* Harvey P. Dale, Aug 18 2021 *)
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
an, p = 1, 2
while True:
yield an
an = next(k for k in count(an+1) if isprime(p*k+1))
p *= an
print(list(islice(agen(), 58))) # Michael S. Branicky, Jan 13 2023
k = 2; a = {}; Do[If[PrimeQ[k n + 1], AppendTo[a, k n+1 ]; k = k n ], {n, 1, 3000}]; a
k = 3; a = {}; Do[If[PrimeQ[k n + 1], AppendTo[a, k n+1 ]; k = k n ], {n, 1, 3000}]; a (*Artur Jasinski*)
4*1+1=5 is prime => a(1)=1. 4*1*2+1=9 is not prime (omitted). 4*1*3+1=13 is prime => a(2)=3.
k = 4; a = {}; Do[If[PrimeQ[k n + 1], k = k n; AppendTo[a, n]], {n, 1, 3000}]; a (* Artur Jasinski *)
k = 5; a = {}; Do[If[PrimeQ[k n + 1], k = k n; AppendTo[a, n]], {n, 1, 3000}]; a
3*1+1=4 is not prime (omitted). a(1)=2 because 3*2+1=7 is prime. a(2)=3 because 3*2*3+1=19 is prime.
k = 3; a = {}; Do[If[PrimeQ[k*n + 1], k = k*n; AppendTo[a, n]], {n, 1, 3000}]; a
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