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A057459 a(n+1) = smallest prime p in the range a(n) < p < a(1)*a(2)*...*a(n) such that p-1 divides a(1)*a(2)*...*a(n); or if no such prime p exists, then a(n+1) = smallest prime > a(n).

%I A057459 #24 Aug 11 2017 03:26:52
%S A057459 2,3,5,7,11,23,31,43,47,67,71,139,211,283,311,331,431,463,659,683,691,
%T A057459 863,947,967,1291,1303,1319,1367,1427,1699,1867,1979,1987,2011,2111,
%U A057459 2131,2311,2531,3011,3083,4099,4423,4643,4691,4831,5171,5179,5683,5839
%N A057459 a(n+1) = smallest prime p in the range a(n) < p < a(1)*a(2)*...*a(n) such that p-1 divides a(1)*a(2)*...*a(n); or if no such prime p exists, then a(n+1) = smallest prime > a(n).
%H A057459 Robert G. Wilson v, <a href="/A057459/b057459.txt">Table of n, a(n) for n = 1..10000</a>
%e A057459 a(3) = 5. Since the product of a(1)*a(2) is 6, there is no prime p < 6 such that p-1 | 6 so the next prime greater than a(2) is 5.
%e A057459 a(9) = 47 since 46 (2*23) | 2*3*5*7*11*23*31*43.
%p A057459 with(numtheory): a:=[2]; P:=1; j:=1;
%p A057459 for n from 2 to 36 do
%p A057459 sw:=-1; P:=P*a[n-1];
%p A057459   for i from j+1 to 1000 do
%p A057459   if (ithprime(i)<P) and (P mod (ithprime(i)-1)) = 0 then Q:=ithprime(i); j:=i; sw:=1; break; fi;
%p A057459                        od:
%p A057459 if sw=-1 then Q:=nextprime(a[n-1]); j:=pi(Q); fi;
%p A057459 a:=[op(a),Q];
%p A057459 od:
%p A057459 a; # _N. J. A. Sloane_, Feb 13 2017
%t A057459 f[s_List] := Block[{b = Times @@ s, p = NextPrime@ Sort[s][[-1]]}, While[ Mod[b, p -1] > 0 && p < b, p = NextPrime@ p]; If[p > b, p = 2; While[ MemberQ[s, p], p = NextPrime@ p]]; Append[s, p]];; Nest[ f, {2}, 50] (* and modified by _Robert G. Wilson v_, Feb 13 2017 *)
%Y A057459 Cf. A007459. See A282027 for another version.
%K A057459 nonn
%O A057459 1,1
%A A057459 _Robert G. Wilson v_, Sep 26 2000