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%I #69 Aug 20 2015 22:53:00
%S 2,5,7,13,17,19,23,29,31,37,41,43,47,53,59,61,67,73,79,83,89,97,101,
%T 103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,193,
%U 197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283
%N a(1) = 2; thereafter, a(n) = smallest prime > a(n-1) and different from gpf(1+Product_{k=1..m}a(k)) for all m < n, where gpf is greatest prime factor.
%C If gpf is replaced by spf, the two sequences diverge at a(40) (here 193, equal to 191 if spf is used instead). - _Giovanni Resta_, Aug 14 2015
%C Among the first 1000 terms, a(n+1) is always the least prime > a(n) except for n = 1, 3, 17, 39, 151 and 422, where a(n+1) = nextprime(a(n)) + 2 except for n = 151 (nextprime = A151800). - _M. F. Hasler_, Aug 17 2015
%C Primes not present in sequence: 3, 11, 71, 191, 911, 2969, 9103, 55871, 313583, ... - _Robert G. Wilson v_, Aug 17 2015
%H Giovanni Resta, <a href="/A261123/b261123.txt">Table of n, a(n) for n = 1..10000</a>
%t gpfQ[x_, p_] := Mod[x, p] == 0 && Block[{n = x / p^IntegerExponent[x, p], f = 2}, While[f < p && n > 1, n /= f^IntegerExponent[n, f]; f = NextPrime@f]; n == 1]; bad[v_, p_] := Catch[Do[If[gpfQ[e + 1, p], Throw@ True], {e, v}]; False]; L = {2}; pr = {2}; While[Length[L] < 100, p = NextPrime@ L[[-1]]; While[bad[pr, p], p = NextPrime@p]; AppendTo[L, p]; AppendTo[pr, p Last@ pr]];L (* _Giovanni Resta_, Aug 14 2015 *)
%o (PARI) vecprod1(v,n)=prod(i=1, n, v[i]);
%o gpf(n)=my(f=factor(n)[, 1]~);f[#f];
%o is_ok(w,n)=my(x=1);for(i=1,#w,if(prime(n)==gpf(1+vecprod1(w,i)),x=0;break));x;
%o get_next(v,p)=until(is_ok(v, p ),p++);p;
%o first(m)={my(v=[],p=1);v=concat(v,prime(1));p++;for(i=2,m,p=get_next(v,p);v=concat(v,prime(p)););v;}
%o (PARI) A261123(N,v=[2],f=[])={while(#v<N,f=setunion(f,factor(prod(i=1,#v,v[i])+1,0)[,1][-1..-1]~);forprime(p=v[#v]+1,,if(!setsearch(f,p),v=concat(v,p);break)));v} \\ _M. F. Hasler_, Aug 17 2015
%Y Cf. A006530, A151800.
%K nonn
%O 1,1
%A _Anders Hellström_, Aug 09 2015
%E a(51)-a(57) from _Giovanni Resta_, Aug 14 2015
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