A054415 Smallest prime factor of n!-1 (for n>2), a(2)=1.

1, 5, 23, 7, 719, 5039, 23, 11, 29, 13, 479001599, 1733, 87178291199, 17, 3041, 19, 59, 653, 124769, 23, 109, 51871, 625793187653, 149, 20431, 29, 239, 31, 265252859812191058636308479999999, 787, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

Offset: 2

Author

Henry Bottomley, May 10 2000

Keywords

• nonn

Comments

The initial term a(2)=1 is not a prime, but it does not affect search results and may prevent submission of duplicates. - M. F. Hasler, Oct 31 2012

Examples

			a(3)=5 because 3!-1=5 which is prime; a(5)=7 because 5!-1=119=7*17 and 7<17
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 2..153 (n = 2..135 from Amiram Eldar)
• 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: n! - 1 (n = 1 to 100); Primorials - 1.
• R. G. Wilson v, Explicit factorizations

Crossrefs

Cf. A002582, A033312, A051301.

Programs

Mathematica

Do[ Print[ FactorInteger[ n! - 1, FactorComplete -> True][ [1, 1] ] ], {n, 3, 32} ]

PARI

A054415(n)=if(n>2,factor(n!-1)[1,1],1)  \\ M. F. Hasler, Oct 31 2012

Formula

Erdős & Stewart show that a(n) > n + (l-o(l))log n/log log n except when n+1 is prime, and that a(n) > n + e(n)sqrt(n) for almost all n where e(n) is any function with lim e(n) = 0. - Charles R Greathouse IV, Dec 05 2012

Extensions

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

More terms from Amiram Eldar, Oct 07 2019