login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002583 Largest prime factor of n! + 1.
(Formerly M0294 N0312)
11
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]

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

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

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: A210598 A330728 A051301 * A068519 A108041 A259254

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 20 16:06 EDT 2020. Contains 337905 sequences. (Running on oeis4.)