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A261123
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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.
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1
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2, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283
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OFFSET
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1,1
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COMMENTS
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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
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
Primes not present in sequence: 3, 11, 71, 191, 911, 2969, 9103, 55871, 313583, ... - Robert G. Wilson v, Aug 17 2015
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(PARI) vecprod1(v, n)=prod(i=1, n, v[i]);
gpf(n)=my(f=factor(n)[, 1]~); f[#f];
is_ok(w, n)=my(x=1); for(i=1, #w, if(prime(n)==gpf(1+vecprod1(w, i)), x=0; break)); x;
get_next(v, p)=until(is_ok(v, p ), p++); p;
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; }
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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