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A002583 Largest prime factor of n! + 1.
(Formerly M0294 N0312)
12
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 (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 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

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

Sequence in context: A330728 A354377 A051301 * A068519 A342848 A339826

Adjacent sequences:  A002580 A002581 A002582 * A002584 A002585 A002586

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

Corrected by Jud McCranie, Jan 03 2001

STATUS

approved

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Last modified September 27 15:39 EDT 2022. Contains 357062 sequences. (Running on oeis4.)