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A084598 a(1) = 2, a(2) = 3; for n >= 2, a(n+1) is smallest prime factor of (Product_{k = 1..n} a(k)) - 1.

%I A084598 #25 Jul 08 2023 11:51:01
%S A084598 2,3,5,29,11,7,13,37,32222189,131,136013303998782209,31,197,19,157,17,
%T A084598 8609,1831129,35977,508326079288931,487,10253,1390043,
%U A084598 18122659735201507243,25319167,9512386441,85577,1031,3650460767,107
%N A084598 a(1) = 2, a(2) = 3; for n >= 2, a(n+1) is smallest prime factor of (Product_{k = 1..n} a(k)) - 1.
%C A084598 Like the Euclid-Mullin sequence A000945, but subtracting rather than adding 1 to the product.
%C A084598 The first 4 terms are identical with A084599. It starts diverging at a(5) because the factorization of 2*3*5*29 - 1 = 869 = 11*79 gives A084598(5)=11 and A084599(5)=79. - _Hugo Pfoertner_, Mar 31 2004
%H A084598 Sean A. Irvine added terms 54 through 61, May 21 2006, giving <a href="/A084598/b084598.txt">Table of n, a(n) for n = 1..61</a>
%H A084598 Dario Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>
%e A084598 a(4) = 29 since 2*3*5 = 30 and 29 is the smallest prime factor of 30-1.
%t A084598 a={2,3}; q=2;
%t A084598 For[n=3,n<=19,n++,
%t A084598     q=q*Last[a];
%t A084598     AppendTo[a,Min[FactorInteger[q-1][[All,1]]]];
%t A084598     ];
%t A084598 a (* _Robert Price_, Jul 17 2015 *)
%Y A084598 Cf. A000945, A005266, A084598, A084599.
%Y A084598 Essentially the same as A005265.
%K A084598 nonn
%O A084598 1,1
%A A084598 _Marc LeBrun_, May 31 2003
%E A084598 More terms from _Hugo Pfoertner_, May 31 2003, using Dario Alpern's ECM