A002583 Largest prime factor of n! + 1.

2, 2, 3, 7, 5, 11, 103, 71, 661, 269, 329891, 39916801, 2834329, 75024347, 3790360487, 46271341, 1059511, 1000357, 123610951, 1713311273363831, 117876683047, 2703875815783, 93799610095769647, 148139754736864591, 765041185860961084291, 38681321803817920159601

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

Examples

			(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
		

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

Crossrefs

Cf. A002582, A002981, A038507, A051301, A056111, A096225.

Programs

Magma

[Maximum(PrimeDivisors(Factorial(n)+1)): n in [0..30]]; // Vincenzo Librandi, Feb 14 2020

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 *)

PARI

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

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

Extensions

More terms from Robert G. Wilson v, Aug 01 2000

Corrected by Jud McCranie, Jan 03 2001