A051301 Smallest prime factor of n!+1.

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

Offset: 0

Author

Labos Elemer

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

Cf. A002583, A002981, A038507, A096225.

Programs

Maple

with(numtheory): A051301 := n -> sort(convert(divisors(n!+1),list))[2]; # Corrected by Peter Luschny, Jul 17 2009

Mathematica

Do[ Print[ FactorInteger[ n! + 1, FactorComplete -> True ] [ [ 1, 1 ] ] ], {n, 0, 38} ]
FactorInteger[#][[1,1]]&/@(Range[0,40]!+1) (* Harvey P. Dale, Oct 16 2021 *)

PARI

a(n)=factor(n!+1)[1,1] \\ Charles R Greathouse IV, Dec 05 2012

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