A002582 -id:A002582 - cp's OEIS frontend

A056110 Highest proper factor of n!-1, or a(n)=1 if n!-1 is not composite.

1, 1, 1, 17, 1, 1, 1753, 32989, 125131, 3070523, 1, 3593203, 1, 76922021647, 6880233439, 18720390952421, 108514808571661, 186286524362683, 19499250680671, 2221345311813453913, 10311933282363373211, 498390560021687969, 991459181683, 104102080827724738147651, 19739193437746837432529

Offset: 2

Henry Bottomley, Jun 12 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 2..135
Hisanori Mishima, Factorizations of many number sequences: n! - 1 (n = 1 to 100).
R. G. Wilson v, Explicit factorizations

Cf. A002582.

pf[n_]:=Module[{c=n!-1},If[PrimeQ[c],1,c/FactorInteger[c][[1,1]]]]; Array[pf,30,2] (* Harvey P. Dale, Dec 13 2012 *)

A056110(n)={n=factor(n!-1);if(norml2(n[,2])>1,factorback(n)/n[1,1],1)} \\ M. F. Hasler, Oct 31 2012

a(n) = A033312(n)/A054415(n)

Edited and extended by M. F. Hasler, Oct 31 2012

A166864 Primes p that divide n! - 1 for some n > 1 other than p-2.

Original entry on oeis.org

17, 23, 29, 31, 53, 59, 61, 67, 71, 73, 83, 89, 97, 103, 107, 109, 137, 139, 149, 151, 167, 193, 199, 211, 223, 227, 233, 239, 251, 271, 277, 283, 307, 311, 331, 359, 379, 389, 397, 401, 419, 431, 439, 449, 457, 461, 463, 467, 479, 487, 499, 503, 521, 547, 557

Offset: 1

Michael B. Porter, Oct 22 2009

nonn

Since n! - 1 = 0 for n=1 and n=2, the restriction n > 1 needed to be placed.
For n >= p, p is one of the factors of n!, so p cannot divide n! - 1.
For n = p-1, by Wilson's Theorem, (p-1)! = -1 (mod p), so p divides (p-1)! + 1, and cannot also divide (p-1)! - 1 unless p = 2.
For n = p-2, again by Wilson's Theorem, (p-1)! = (p-1)(p-2)! = (-1)(p-2)! = -1 (mod p), so (p-2)! = 1 (mod p) and p divides (p-2)! - 1. As a result, only 2 <= n <= p-3 needs to be searched.

			17 is included in the sequence since 17 divides 5! - 1 = 119.
19 is not included in the sequence since the only n for which 19 divides n! - 1 is n = 17.

Eric Weisstein's World of Mathematics, Wilson's Theorem

Cf. A000142, A002582, A033312, A054415.

isA166864(n) = {local(r);r=0;for(i=2,n-3,if((i!-1)%n==0,r=1));r}

cp's OEIS Frontend

A056110 Highest proper factor of n!-1, or a(n)=1 if n!-1 is not composite.

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