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 A051301 Smallest prime factor of n!+1. 12
 2, 2, 3, 7, 5, 11, 7, 71, 61, 19, 11, 39916801, 13, 83, 23, 59, 17, 661, 19, 71, 20639383, 43, 23, 47, 811, 401, 1697, 10888869450418352160768000001, 29, 14557, 31, 257, 2281, 67, 67411, 137, 37, 13763753091226345046315979581580902400000001 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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 if and only if p is a prime. If n is in A002981, then a(n) = n!+1. - Chai Wah Wu, Jul 15 2019 REFERENCES Albert H. Beiler, "Recreations in The Theory of Numbers, The Queen of Mathematics Entertains," Dover Publ. NY, 1966, Page 49. M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). LINKS Chai Wah Wu, Table of n, a(n) for n = 0..138 n = 0..100 derived from Hisanori Mishima's data by T. D. Noe. A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570. P. Erdős and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519. M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy] Hisanori Mishima, Factorizations of many number sequences Hisanori Mishima, Factorizations of many number sequences R. G. Wilson v, Explicit factorizations FORMULA 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 By Wilson's theorem, a(n) >=  n + 1 with equality if and only if n + 1 is prime. - Chai Wah Wu, Jul 14 2019 EXAMPLE a(3) = 7 because 3! + 1 = 7. a(4) = 5 because 4! + 1 = 25 = 5^2. (5! + 1 is also the square of a prime). a(6) = 7 because 6! + 1 = 721 = 7 * 103. MAPLE with(numtheory): A051301 := n -> sort(convert(divisors(n!+1), list)); # Corrected by Peter Luschny, Jul 17 2009 MATHEMATICA Do[ Print[ FactorInteger[ n! + 1, FactorComplete -> True ] [ [ 1, 1 ] ] ], {n, 0, 38} ] PROG (PARI) a(n)=factor(n!+1)[1, 1] \\ Charles R Greathouse IV, Dec 05 2012 CROSSREFS Cf. A002583, A002981, A038507, A096225. Sequence in context: A267822 A210598 A330728 * A002583 A068519 A108041 Adjacent sequences:  A051298 A051299 A051300 * A051302 A051303 A051304 KEYWORD nonn AUTHOR STATUS approved

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Last modified July 6 23:01 EDT 2020. Contains 335484 sequences. (Running on oeis4.)