

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
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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  (p1)! + 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), 2426 (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), 567570.
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. 513519.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 2426 (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 + (lo(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} ]


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

Labos Elemer


STATUS

approved



