A054415 -id:A054415 - cp's OEIS frontend

A033312 a(n) = n! - 1.

0, 0, 1, 5, 23, 119, 719, 5039, 40319, 362879, 3628799, 39916799, 479001599, 6227020799, 87178291199, 1307674367999, 20922789887999, 355687428095999, 6402373705727999, 121645100408831999, 2432902008176639999, 51090942171709439999, 1124000727777607679999

Offset: 0

N. J. A. Sloane. This sequence appeared in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Eric W. Weisstein. Entry revised by N. J. A. Sloane, Jun 12 2012

a(n) gives the index number in any table of permutations of the entry in which the last n + 1 items are reversed. - Eugene McDonnell (eemcd(AT)mac.com), Dec 03 2004
a(n), n >= 1, has the factorial representation [n - 1, n - 2, ..., 1, 0]. The (unique) factorial representation of a number m from {0, 1, ... n! - 1} is m = sum(m_j(n)*j!, j = 0 .. n - 1) with m_j(n) from {0, 1, .., j}, n>=1. This is encoded as [m_{n-1},m_{n-2},...,m+1,m_0] with m_0=0. This can be interpreted as (D. N.) Lehmer code for the lexicographic rank of permutations of the symmetric group S_n (see the W. Lang link under A136663). The Lehmer code [n - 1, n - 2, ..., 1, 0] stands for the permutation [n, n - 1, ..., 1] (the last in lexicographic order). - Wolfdieter Lang, May 21 2008
For n >= 3: a(n) = numbers m for which there is one iteration {floor (r / k)} for k = n, n - 1, n - 2, ... 2 with property r mod k = k - 1 starting at r = m. For n = 5: a(5) = 119; floor (119 / 5) = 23, 119 mod 5 = 4; floor (23 / 4) = 5, 23 mod 4 = 3; floor (5 / 3) = 1, 5 mod 3 = 2; floor (1 / 2) = 0; 1 mod 2 = 1. - Jaroslav Krizek, Jan 23 2010
For n = 4, define the sum of all possible products of 1, 2, 3, 4 to be 1 + 2 + 3 + 4 add 1*2 + 1*3 + 1*4 add 2*3 + 2*4 + 3*4 add 1*2*3 + 1*2*4 + 1*3*4 + 2*3*4 add 1*2*3*4. The sum of this is 119 = (4 + 1)! - 1. For n = 5 I get the sum 719 = (5 + 1)! - 1. The proof for the general case seems to follow by induction. - J. M. Bergot, Jan 10 2011

			G.f. = x^2 + 5*x^3 + 23*x^4 + 119*x^5 + 719*x^6 + 5039*x^7 + 40319*x^8 + ...

Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identity 181, p. 92.
Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 6, 1969, p. 3, 1993.
Problem 598, J. Rec. Math., 11 (1978), 68-69.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
The IMO Compendium, Problem 6, 1st Canadian Mathematical Olympiad 1969.
Stéphane Legendre and Philippe Paclet, On the Permutations Generated by Cyclic Shift , J. Int. Seq. 14 (2011) # 11.3.2.
Gerard P. Michon, Wilson's Theorem.
Hisanori Mishima, Factorizations of many number sequences.
Hisanori Mishima, Factorizations of many number sequences.
Michael Penn, Make it look like a simple calculus problem., YouTube video, 2021.
Andrew Walker, Factors of n! +- 1.
Eric Weisstein's World of Mathematics, Factorial.
Eric Weisstein's World of Mathematics, Permutation Pattern.
Index entries for sequences related to factorial base representation.
Index entries for sequences related to factorial numbers.
Index to sequences related to Olympiads.

Cf. A000142, A001563 (first differences), A002582, A002982, A038507 (factorizations), A054415, A056110, A331373.

Row sums of A008291.

[Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

FoldList[#1*#2 + #2 - 1 &, 0, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
Range[0, 19]! - 1 (* Alonso del Arte, Jan 24 2013 *)

A033312(n):= n!-1$
makelist(A033312(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=n!-1 \\ Charles R Greathouse IV, Jul 19 2011

a(n) = Sum_{k = 1 .. n} (k-1)*(k-1)!.
a(n) = a(n - 1)*(n - 1) + a(n - 1) + n - 1, a(0) = 0. - Reinhard Zumkeller, Feb 03 2003
a(0) = a(1) = 0, a(n) = a(n - 1) * n + (n - 1) for n >= 2. - Jaroslav Krizek, Jan 23 2010
E.g.f.: 1/(1 - x) - exp(x). - Sergei N. Gladkovskii, Jun 29 2012
0 = 1 + a(n)*(+a(n+1) - a(n+2)) + a(n+1)*(+3 + a(n+1)) + a(n+2)*(-1) for n>=0. - Michael Somos, Feb 24 2017
Sum_{n>=2} 1/a(n) = A331373. - Amiram Eldar, Nov 11 2020

A002582 Largest prime factor of n! - 1.

Original entry on oeis.org

1, 5, 23, 17, 719, 5039, 1753, 2999, 125131, 7853, 479001599, 3593203, 87178291199, 1510259, 6880233439, 256443711677, 478749547, 78143369, 19499250680671, 4826713612027, 170006681813, 498390560021687969, 991459181683, 114776274341482621993

Offset: 2

N. J. A. Sloane

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).

Sean A. Irvine, Table of n, a(n) for n = 2..135
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
R. G. Wilson v, Explicit factorizations
J. W. Wrench, Jr., Letters to N. J. A. Sloane, Feb 1974

Cf. A002583, A033312, A054415, A056110.

[1] cat [Maximum(PrimeDivisors(Factorial(n)-1)): n in [3..30]]; // Vincenzo Librandi, Feb 14 2020

Table[FactorInteger[n! - 1][[-1, 1]], {n, 2, 25}] (* Harvey P. Dale, Aug 29 2011 *)

a(n)=if(n>2,my(f=factor(n!-1)[,1]);f[#f],1) \\ Charles R Greathouse IV, Dec 05 2012

Erdős & Stewart show that a(n) > n + (l-o(l))log n/log log n and lim sup a(n)/n > 2. - Charles R Greathouse IV, Dec 05 2012

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

A104358 Smallest prime factor of A104357(n) = A104350(n) - 1.

Original entry on oeis.org

1, 5, 11, 59, 179, 1259, 11, 7559, 37799, 415799, 71, 227, 5981, 9067, 1135133999, 11717, 61, 79, 5499724229999, 97, 1543, 31, 29220034833989999, 8937119, 181, 401, 124759, 443851, 31, 2141, 3082663, 8191, 37230797, 1697, 1408101540804746673385499999, 10613, 73, 59, 107, 79, 617, 163, 173

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

a(n) = A020639(A104357(n)).

Tyler Busby, Table of n, a(n) for n = 2..181 (terms 2..120 from Chai Wah Wu, terms 121..160 from Max Alekseyev)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A104357, A104359, A104360, A104361, A104362, A104363, A104364, A104366, A054415, A057713.

Typo in data corrected by Gionata Neri, Oct 20 2017

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

Original entry on oeis.org

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

A033312 a(n) = n! - 1.

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A002582 Largest prime factor of n! - 1.

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A104358 Smallest prime factor of A104357(n) = A104350(n) - 1.

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

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

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PARI