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

0,1

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))[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 *)

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: A210598 A330728 A354377 * A002583 A068519 A342848

Adjacent sequences: A051298 A051299 A051300 * A051302 A051303 A051304

Keyword

nonn

Author

Labos Elemer

Status

approved