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A144717 a(n) = smallest positive integer > a(n-1) such that 2*a(1)*a(2)*...*a(n) + 1 is prime.

%I A144717 #29 Jan 13 2023 08:41:23
%S A144717 1,2,3,5,7,8,9,11,12,14,17,20,24,30,34,44,72,85,86,92,115,122,125,132,
%T A144717 142,150,161,162,181,186,198,224,248,252,282,283,290,307,319,321,344,
%U A144717 350,376,445,476,567,623,676,682,704,741,749,786,803,806,893,1014,1046
%N A144717 a(n) = smallest positive integer > a(n-1) such that 2*a(1)*a(2)*...*a(n) + 1 is prime.
%H A144717 Jon E. Schoenfield, <a href="/A144717/b144717.txt">Table of n, a(n) for n = 1..505</a> (lists all terms < 10^5)
%e A144717 a(1)=1 because a(0) is not defined and 2*1 + 1 = 3 is prime;
%e A144717 a(2)=2 because 2*1*2 + 1 = 5 is prime;
%e A144717 a(3)=3 because 2*1*2*3 + 1 = 13 is prime;
%e A144717 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.
%t A144717 k = 2; a = {}; Do[If[PrimeQ[k n + 1], k = k n; AppendTo[a, n]], {n, 1, 3000}]; a (* _Artur Jasinski_ *)
%t A144717 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 *)
%o A144717 (Python)
%o A144717 from sympy import isprime
%o A144717 from itertools import count, islice
%o A144717 def agen(): # generator of terms
%o A144717     an, p = 1, 2
%o A144717     while True:
%o A144717         yield an
%o A144717         an = next(k for k in count(an+1) if isprime(p*k+1))
%o A144717         p *= an
%o A144717 print(list(islice(agen(), 58))) # _Michael S. Branicky_, Jan 13 2023
%Y A144717 Cf. A046966, A046972, A144718, A144722, A144723, A144724, A144725, A144726, A144727, A144728, A144729, A144730, A144731.
%K A144717 nonn,nice
%O A144717 1,2
%A A144717 _Artur Jasinski_, Sep 19 2008
%E A144717 Edited by _N. J. A. Sloane_, Sep 21 2017 following suggestions from Richard C. Schroeppel