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A002583
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Largest prime factor of n! + 1.
(Formerly M0294 N0312)
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13
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2, 2, 3, 7, 5, 11, 103, 71, 661, 269, 329891, 39916801, 2834329, 75024347, 3790360487, 46271341, 1059511, 1000357, 123610951, 1713311273363831, 117876683047, 2703875815783, 93799610095769647, 148139754736864591, 765041185860961084291, 38681321803817920159601
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OFFSET
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0,1
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COMMENTS
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Theorem: For any N, there is a prime > N. Proof: Consider any prime factor of N!+1.
Cf. Wilson's theorem (1770): p | (p-1)! + 1 iff p is a prime.
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REFERENCES
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M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Blake C. Stacey, Equiangular Lines, Ch. 1, A First Course in the Sporadic SICs, SpringerBriefs in Math. Phys. (2021) Vol. 41, see page 5.
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FORMULA
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Erdős & Stewart show that a(n) > n + (1-o(1))log n/log log n and lim sup a(n)/n > 2. - Charles R Greathouse IV, Dec 05 2012
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EXAMPLE
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(0!+1)=[2], (1!+1)=[2], (2!+1)=[3], (3!+1)=[7], (4!+1)=25=5*[5], (5!+1)=121=11*[11], (6!+1)=721=7*[103], (7!+1)=5041=71*[71], etc. - Mitch Cervinka (puritan(AT)toast.net), May 11 2009
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MATHEMATICA
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PrimeFactors[n_]:=Flatten[Table[ #[[1]], {1}]&/@FactorInteger[n]]; Table[PrimeFactors[n!+1][[ -1]], {n, 0, 35}] ..and/or.. Table[FactorInteger[n!+1, FactorComplete->True][[ -1, 1]], {n, 0, 35}] (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
FactorInteger[#][[-1, 1]]&/@(Range[0, 30]!+1) (* Harvey P. Dale, Sep 04 2017 *)
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PROG
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(Magma) [Maximum(PrimeDivisors(Factorial(n)+1)): n in [0..30]]; // Vincenzo Librandi, Feb 14 2020
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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