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A005266 a(1)=3, b(n) = Product_{k=1..n} a(k), a(n+1) is the largest prime factor of (b(n)-1).

%I A005266 M2247 #38 Jan 31 2022 03:06:40
%S A005266 3,2,5,29,79,68729,3739,6221191,157170297801581,
%T A005266 70724343608203457341903,46316297682014731387158877659877,
%U A005266 78592684042614093322289223662773,181891012640244955605725966274974474087,547275580337664165337990140111772164867508038795347198579326533639132704344301831464707648235639448747816483406685904347568344407941
%N A005266 a(1)=3, b(n) = Product_{k=1..n} a(k), a(n+1) is the largest prime factor of (b(n)-1).
%C A005266 Suggested by Euclid's proof that there are infinitely many primes.
%C A005266 a(15) requires completing the factorization: 13 * 67 * 14479 * 167197 * 924769 * 2688244927 * 888838110930755119 * 14372541055015356634061816579965403 * C211 where C211=6609133306626483634448666494646737799624640616060730302142187545405582531010390290502001156883917023202671554510633460047901459959959325342475132778791495112937562941066523907603281586796876335607258627832127303. - _Sean A. Irvine_, Nov 10 2009
%D A005266 R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
%D A005266 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005266 S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32.
%H A005266 R. K. Guy and R. Nowakowski, <a href="/A000945/a000945_5.pdf">Discovering primes with Euclid</a>, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
%H A005266 Des MacHale, <a href="http://dx.doi.org/10.2307/3621650">Infinitely many proofs that there are infinitely many primes</a>, Math. Gazette, 97 (No. 540, 2013), 495-498.
%H A005266 S. S. Wagstaff, Jr., <a href="/A000945/a000945_4.pdf">Computing Euclid's primes</a>, Bull. Institute Combin. Applications, 8 (1993), 23-32. (Annotated scanned copy)
%H A005266 <a href="http://factordb.com/index.php?id=1100000000216023476">Status of a(15) in factordb.com</a>.
%p A005266 with(numtheory):
%p A005266 a:= proc(n) option remember;
%p A005266       `if`(n=1, 3, max(factorset(mul(a(i), i=1..n-1)-1)[]))
%p A005266     end:
%p A005266 seq(a(n), n=1..10);  # _Alois P. Heinz_, Sep 26 2013
%t A005266 a[0] = 3; a[n_] := a[n] = Block[{p = Times @@ (a[#] & /@ Range[0, n - 1]) - 1}, FactorInteger[p][[-1, 1]]]; Array[a, 13] (* _Robert G. Wilson v_, Sep 26 2013 *)
%Y A005266 Cf. A000945, A000946, A005265.
%Y A005266 Essentially the same as A084599.
%K A005266 nonn,nice,hard
%O A005266 1,1
%A A005266 _N. J. A. Sloane_
%E A005266 a(14) from Joe K. Crump (joecr(AT)carolina.rr.com), Jul 26 2000