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%I #36 Oct 16 2021 19:46:08
%S 2,2,3,7,5,11,7,71,61,19,11,39916801,13,83,23,59,17,661,19,71,
%T 20639383,43,23,47,811,401,1697,10888869450418352160768000001,29,
%U 14557,31,257,2281,67,67411,137,37,13763753091226345046315979581580902400000001
%N Smallest prime factor of n!+1.
%C Theorem: For any N, there is a prime > N. Proof: Consider any prime factor of N! + 1.
%C Cf. Wilson's Theorem (1770): p | (p-1)! + 1 if and only if p is a prime.
%C If n is in A002981, then a(n) = n!+1. - _Chai Wah Wu_, Jul 15 2019
%D Albert H. Beiler, "Recreations in The Theory of Numbers, The Queen of Mathematics Entertains," Dover Publ. NY, 1966, Page 49.
%D M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
%H Chai Wah Wu, <a href="/A051301/b051301.txt">Table of n, a(n) for n = 0..138</a> n = 0..100 derived from Hisanori Mishima's data by T. D. Noe.
%H A. Borning, <a href="http://dx.doi.org/10.1090/S0025-5718-1972-0308018-5 ">Some results for k!+-1 and 2.3.5...p+-1</a>, Math. Comp., 26 (1972), 567-570.
%H P. Erdős and C. L. Stewart, <a href="http://www.renyi.hu/~p_erdos/1976-27.pdf">On the greatest and least prime factors of n! + 1</a>, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519.
%H M. Kraitchik, <a href="/A002582/a002582.pdf">On the divisibility of factorials</a>, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha104.htm">Factorizations of many number sequences</a>
%H R. G. Wilson v, <a href="/A038507/a038507.txt">Explicit factorizations</a>
%F Erdős & Stewart show that a(n) > n + (l-o(l))log n/log log n except when n + 1 is prime, and that a(n) > n + e(n)sqrt(n) for almost all n where e(n) is any function with lim e(n) = 0. - _Charles R Greathouse IV_, Dec 05 2012
%F By Wilson's theorem, a(n) >= n + 1 with equality if and only if n + 1 is prime. - _Chai Wah Wu_, Jul 14 2019
%e a(3) = 7 because 3! + 1 = 7.
%e a(4) = 5 because 4! + 1 = 25 = 5^2. (5! + 1 is also the square of a prime).
%e a(6) = 7 because 6! + 1 = 721 = 7 * 103.
%p with(numtheory): A051301 := n -> sort(convert(divisors(n!+1),list))[2]; # Corrected by _Peter Luschny_, Jul 17 2009
%t Do[ Print[ FactorInteger[ n! + 1, FactorComplete -> True ] [ [ 1, 1 ] ] ], {n, 0, 38} ]
%t FactorInteger[#][[1,1]]&/@(Range[0,40]!+1) (* _Harvey P. Dale_, Oct 16 2021 *)
%o (PARI) a(n)=factor(n!+1)[1,1] \\ _Charles R Greathouse IV_, Dec 05 2012
%Y Cf. A002583, A002981, A038507, A096225.
%K nonn
%O 0,1
%A _Labos Elemer_
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