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 iff p is a prime.
If n is in A002981, then a(n) = n!+1. - Chai Wah Wu, Jul 15 2019
REFERENCES
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Georg Fischer, Table of n, a(n) for n = 0..139 (first 101 terms originally 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]
Li Lai, On the largest prime divisor of n! + 1, arXiv:2103.14894 [math.NT], 2021.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
Blake C. Stacey, Equiangular Lines, Ch. 1, A First Course in the Sporadic SICs, SpringerBriefs in Math. Phys. (2021) Vol. 41, see page 5.
R. G. Wilson v, Explicit factorizations
FORMULA
Erdős & Stewart show that a(n) > n + (1-o(1))log n/log log n and lim sup a(n)/n > 2. - Charles R Greathouse IV, Dec 05 2012
Lai proves that lim sup a(n)/n > 7.238. - Charles R Greathouse IV, Jun 22 2021
EXAMPLE
(0!+1)=[2], (1!+1)=[2], (2!+1)=[3], (3!+1)=[7], (4!+1)=25=5*[5], (5!+1)=121=11*[11], (6!+1)=721=7*[103], (7!+1)=5041=71*[71], etc. - Mitch Cervinka (puritan(AT)toast.net), May 11 2009
MATHEMATICA
PrimeFactors[n_]:=Flatten[Table[ #[[1]], {1}]&/@FactorInteger[n]]; Table[PrimeFactors[n!+1][[ -1]], {n, 0, 35}] ..and/or.. Table[FactorInteger[n!+1, FactorComplete->True][[ -1, 1]], {n, 0, 35}] (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
FactorInteger[#][[-1, 1]]&/@(Range[0, 30]!+1) (* Harvey P. Dale, Sep 04 2017 *)
PROG
(PARI) a(n)=my(f=factor(n!+1)[, 1]); f[#f] \\ Charles R Greathouse IV, Dec 05 2012
(Magma) [Maximum(PrimeDivisors(Factorial(n)+1)): n in [0..30]]; // Vincenzo Librandi, Feb 14 2020
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Aug 01 2000
Corrected by Jud McCranie, Jan 03 2001
STATUS
approved