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A007996
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Primes that divide at least one term of the sequence f given by f(1) = 2, f(n+1) = n^2-n+1.
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6
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2, 3, 7, 13, 43, 73, 139, 181, 547, 607, 1033, 1171, 1459, 1861, 1987, 2029, 2287, 2437, 4219, 4519, 6469, 7603, 8221, 9829, 12763, 13147, 13291, 13999, 15373, 17881, 17977, 19597, 20161, 20479, 20641, 20857, 20929, 21661, 23689, 23773, 27031
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Or, let S_1 = [2] and let S_{n+1} = list formed by sorting the union of S_n together with all prime factors of 1 + Product_i S_n(i) into increasing order; sequence is limit as n -> infinity of S_n.
Prime divisors of the terms of Sylvester's sequence A000058. - Max Alekseyev (maxale(AT)gmail.com), Jan 03 2004. Also of A007018. - N. J. A. Sloane (njas(AT)research.att.com), Jan 27 2007
Because all terms of the sequence f(n) are coprime, a prime can divide at most one term. Odoni shows that primes p>3 in this sequence must satisfy p=1 (mod 6). [From T. D. Noe (noe(AT)sspectra.com), Sep 25 2010]
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REFERENCES
| R. W. K. Odoni, On the prime divisors of the sequence w_{n+1} = 1+w_1 ... w_n, J. London Math. Soc. 32 (1985), 1-11.
Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Dec 2006
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..8181 (all primes < 2^32, from Andersen)
Eric Weisstein's World of Mathematics, Sylvester's sequence
Jens Kruse Andersen, Factorization of Sylvester's sequence
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MAPLE
| n := 1; for p do if isprime(p) then x := 2 mod p; S := {}; while not member(x, S) do if x=0 then a[n] := p; n := n+1; break; fi; S := S union {x}; x := (x^2-x+1) mod p; od; fi; od;
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MATHEMATICA
| t={}; p=1; While[Length[t]<100, p=NextPrime[p]; s=Mod[2, p]; k=0; modSet={}; While[s>0 && !MemberQ[modSet, s], AppendTo[modSet, s]; k++; s=Mod[s^2-s+1, p]]; If[s==0, AppendTo[t, {p, k}]]]; Transpose[t][[1]] [From T. D. Noe (noe(AT)sspectra.com), Sep 25 2010]
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CROSSREFS
| The missing primes form A096264.
Cf. A000058, A014546, A091335, A091336.
Cf. A180871 (k such that a(n) divides A000058(k)) [From T. D. Noe (noe(AT)sspectra.com), Sep 25 2010]
Sequence in context: A078749 A046062 A096263 * A085872 A075059 A070858
Adjacent sequences: A007993 A007994 A007995 * A007997 A007998 A007999
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KEYWORD
| nonn
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AUTHOR
| Bennett Battaile (bennett.battaile(AT)autodesk.com)
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EXTENSIONS
| More terms from Max Alekseyev (maxale(AT)gmail.com), Jan 03 2004
Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jan 28 2007
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