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A002583 Largest prime factor of n! + 1.
(Formerly M0294 N0312)
10
2, 2, 3, 7, 5, 11, 103, 71, 661, 269, 329891, 39916801, 2834329, 75024347, 3790360487, 46271341, 1059511, 1000357, 123610951, 1713311273363831, 117876683047, 2703875815783, 93799610095769647, 148139754736864591 (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.

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

T. D. Noe, Table of n, a(n) for n = 0..100 (derived from Hisanori Mishima's data)

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A267822 A210598 A051301 * A068519 A083702 A108041

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 May 26 18:58 EDT 2017. Contains 287129 sequences.