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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.

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

T. D. Noe, Table of n, a(n) for n = 0..100 (derived from Hisanori Mishima's data)

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

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} ]

PROG

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

CROSSREFS

Cf. A002583, A038507, A096225.

Sequence in context: A209746 A267822 A210598 * A002583 A068519 A108041

Adjacent sequences:  A051298 A051299 A051300 * A051302 A051303 A051304

KEYWORD

nonn

AUTHOR

Labos Elemer

STATUS

approved

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Last modified November 16 04:54 EST 2018. Contains 317257 sequences. (Running on oeis4.)