A002583 -id:A002583 - cp's OEIS frontend

A038507 a(n) = n! + 1.

2, 2, 3, 7, 25, 121, 721, 5041, 40321, 362881, 3628801, 39916801, 479001601, 6227020801, 87178291201, 1307674368001, 20922789888001, 355687428096001, 6402373705728001, 121645100408832001, 2432902008176640001, 51090942171709440001, 1124000727777607680001, 25852016738884976640001

Offset: 0

N. J. A. Sloane

"For n = 4, 5 and 7, n!+1 is a square. Sierpiński asked if there are any other values of n with this property." p. 82 of Ogilvy and Anderson (see A146968).
Number of {12,12*,1*2,21*,2*1}-avoiding signed permutations in the hyperoctahedral group.
After Wilson's Theorem: if (n+1) is prime then (n+1) is the smallest prime factor of a(n). - Karl-Heinz Hofmann, Aug 21 2024

			G.f. = 2 + 2*x + 3*x^2 + 7*x^3 + 25*x^4 + 121*x^5 + 721*x^6 + 5041*x^7 + ...

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, p. 82.
Wacław Sierpiński, On some unsolved problems of arithmetics, Scripta Mathematica, vol. 25 (1960), p. 125.
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
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 763 and Encyclopedia of Combinatorial Structures 834
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Romeo Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012
Gerard P. Michon, Wilson's Theorem
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Andrew Walker, Factors of n! +- 1
Arthur T. White, Ringing the changes, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 203-215.
Robert G. Wilson v, Explicit factorizations
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
Index entries for sequences related to factorial numbers

Cf. A000142, A002583, A002981, A033312, A051301, A056111, A146968, A217702.

a038507 = (+ 1) . a000142
a038507_list = 2 : f 1 2 where
   f x y = z : f (x + 1) z where z = x * (y - 1) + 1
-- Reinhard Zumkeller, Mar 20 2013

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

Range[0,20]!+1 (* Harvey P. Dale, May 06 2012 *)

A038507(n):= n!+1$
makelist(A038507(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=n!+1 \\ Charles R Greathouse IV, Nov 20 2012

from math import factorial
def A038507(n): return factorial(n) + 1 # Karl-Heinz Hofmann, Aug 21 2024

[factorial(n) + 1 for n in range(0,24)] # Stefano Spezia, Apr 21 2025

a(n) = n * (a(n-1) - 1) + 1. - Reinhard Zumkeller, Mar 20 2013
0 = a(n)*(a(n+1) - 5*a(n+2) + 5*a(n+3) - a(n+4)) + a(n+1)*(a(n+1) + a(n+2) - 6*a(n+3) + 2*a(n+4)) + a(n+2)*(3*a(n+2) - a(n+3) - a(n+4)) + a(n+3)*(a(n+3)) if n>=0. - Michael Somos, Apr 23 2014
From Ilya Gutkovskiy, Jan 20 2017: (Start)
E.g.f: exp(x) + 1/(1 - x).
Sum_{n>=0} 1/a(n) = A217702. (End)

Additional comments from Jason Earls, Apr 01 2001
Numericana.com URL fixed by Gerard P. Michon, Mar 30 2010
Entry revised by N. J. A. Sloane, Jun 10 2012

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

A051301 Smallest prime factor of n!+1.

Original entry on oeis.org

2, 2, 3, 7, 5, 11, 7, 71, 61, 19, 11, 39916801, 13, 83, 23, 59, 17, 661, 19, 71, 20639383, 43, 23, 47, 811, 401, 1697, 10888869450418352160768000001, 29, 14557, 31, 257, 2281, 67, 67411, 137, 37, 13763753091226345046315979581580902400000001

Offset: 0

Labos Elemer

nonn

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 if and only if p is a prime.
If n is in A002981, then a(n) = n!+1. - Chai Wah Wu, Jul 15 2019

			a(3) = 7 because 3! + 1 = 7.
a(4) = 5 because 4! + 1 = 25 = 5^2. (5! + 1 is also the square of a prime).
a(6) = 7 because 6! + 1 = 721 = 7 * 103.

Albert H. Beiler, "Recreations in The Theory of Numbers, The Queen of Mathematics Entertains," Dover Publ. NY, 1966, Page 49.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).

Chai Wah Wu, Table of n, a(n) for n = 0..138 n = 0..100 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
R. G. Wilson v, Explicit factorizations

Cf. A002583, A002981, A038507, A096225.

with(numtheory): A051301 := n -> sort(convert(divisors(n!+1),list))[2]; # Corrected by Peter Luschny, Jul 17 2009

Do[ Print[ FactorInteger[ n! + 1, FactorComplete -> True ] [ [ 1, 1 ] ] ], {n, 0, 38} ]
FactorInteger[#][[1,1]]&/@(Range[0,40]!+1) (* Harvey P. Dale, Oct 16 2021 *)

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

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

By Wilson's theorem, a(n) >= n + 1 with equality if and only if n + 1 is prime. - Chai Wah Wu, Jul 14 2019

A104367 Greatest prime factor of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 97, 2521, 7561, 367, 415801, 1247401, 97103, 594311, 2689891, 269, 415147, 1434493, 1099944846001, 13421, 938977307561, 1687166397251, 6121943187511, 13027211250107, 146100174169950001, 1389833, 10603380543703, 2129284819, 1156675078903494150001, 132597517693, 47172675889, 11159737, 20350106034371

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..158 (terms 1..76 from Amiram Eldar, terms 142..151 from Tyler Busby)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A006530, A002583, A002585, A104350, A104359, A104365, A104366, A104368, A104369, A104370, A104371, A104372.

a[n_] := FactorInteger[1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]][[-1, 1]]; Array[a, 76] (* Amiram Eldar, Feb 12 2020 *)

a(n) = A006530(A104365(n)).

Corrected by T. D. Noe, Nov 15 2006

A064144 a(n) is the number of divisors of n!+1.

Original entry on oeis.org

2, 2, 2, 3, 3, 4, 3, 4, 8, 4, 2, 6, 4, 4, 8, 32, 8, 64, 4, 4, 8, 8, 12, 4, 4, 4, 2, 4, 8, 32, 16, 16, 32, 4, 32, 64, 2, 4, 16, 128, 2, 8, 16, 8, 8, 8, 16, 4, 32, 32, 64, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 8, 32, 8

Offset: 1

Vladeta Jovovic, Sep 11 2001

nonn

Max Alekseyev, Table of n, a(n) for n = 1..139
William Gerst, A conjecture on the prime factorization of n!+1, arXiv:1809.07360 [math.GM], 2018.
Hisanori Mishima, Factorizations of many number sequences

Cf. A000005, A002583, A027423, A002981, A002982, A064145, A078778, A181764.

Do[ Print[ DivisorSigma[0, n! + 1]], {n, 1, 40} ]

a(n) = numdiv(n! + 1); \\ Harry J. Smith, Sep 09 2009

from math import factorial
from sympy import divisor_count
def A064144(n): return divisor_count(factorial(n)+1) # Chai Wah Wu, Oct 20 2023

a(n) = tau(n!+1).

More terms from Robert G. Wilson v, Oct 04 2001
a(42)-a(64) from Harry J. Smith, Sep 09 2009
Edited by Jon E. Schoenfield, Jun 21 2018

A164314 Largest prime factor of n^2 - 2.

Original entry on oeis.org

2, 7, 7, 23, 17, 47, 31, 79, 7, 17, 71, 167, 97, 223, 127, 41, 23, 359, 199, 439, 241, 31, 41, 89, 337, 727, 23, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 31, 47, 73, 881, 1847, 967, 17, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 47, 3023, 1567, 191

Offset: 2

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A002583, A014442, A069902.

a:= n-> max(numtheory[factorset](n^2-2)):
seq(a(n), n=2..60);  # Alois P. Heinz, Jul 22 2017

Table[FactorInteger[n^2 - 2][[-1, 1]], {n, 2, 57}] (* Michael De Vlieger, Jul 22 2017 *)

a(n) = vecmax(factor(n^2-2)[,1]); \\ Michel Marcus, Jul 22 2017

a(n) = A006530(A008865(n)).

Offset corrected by R. J. Mathar, Aug 21 2009

A056111 Highest proper factor of n!+1.

Original entry on oeis.org

1, 1, 1, 1, 5, 11, 103, 71, 661, 19099, 329891, 1, 36846277, 75024347, 3790360487, 22163972339, 1230752346353, 538105034941, 336967037143579, 1713311273363831, 117876683047, 1188161445853707907, 48869596859895986087, 550042909337978226383, 765041185860961084291

Offset: 0

Henry Bottomley, Jun 12 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 0..139
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A002583, A038507, A051301, A056110.

Mathematica
```
a[n_] := Divisors[n!+1][[ -2]]
```

a(n) = A038507(n)/A051301(n).

Corrected and extended by Dean Hickerson, Aug 30 2001

More terms from Amiram Eldar, Oct 07 2019

A096225 a(0) = 1; for n >= 0, a(n+1) = smallest prime factor of a(n)! + 1.

Original entry on oeis.org

1, 2, 3, 7, 71, 6653, 25469, 15750503

Offset: 0

N. J. A. Sloane, Aug 09 2004

			71!+1 is the product of 6653 and a large prime.

Cf. A002583, A051301.

a[1] = 2; a[n_] := Block[{p = PrimePi[a[n - 1]] + 1, r = a[n - 1]! + 1}, While[ Mod[r, Prime[p]] != 0, p++ ]; Prime[p]]; Do[ Print[ a[n]], {n, 7}] (* Robert G. Wilson v, Aug 12 2004 *)
NestList[FactorInteger[#!+1][[1,1]]&,1,7] (* Harvey P. Dale, Sep 20 2016 *)

a(6) and a(7) from Robert G. Wilson v, Aug 12 2004

A362779 Triangular array read by rows: T(n,k) is the greatest prime factor of n!*k + 1, n >= 1, 1 <= k <= n.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 5, 7, 73, 97, 11, 241, 19, 37, 601, 103, 131, 2161, 67, 277, 149, 71, 593, 15121, 20161, 79, 30241, 35281, 661, 7331, 1657, 161281, 449, 241921, 282241, 6863, 269, 2477, 1088641, 1451521, 78887, 2177281, 5281, 2903041, 192113, 329891, 29383, 10886401, 62297, 18144001, 2243, 251501, 29030401, 32659201, 843907

Offset: 1

Joe B. Stephen, May 03 2023

The primes in each row are distinct because n!*k + 1 are coprime for 1 <= k <= n, and hence this array gives a simple proof that there are infinitely many prime numbers.

			Triangle T(n,k) begins:
  n\k   1    2    3    4    5    6 ...
  1     2
  2     3    5
  3     7   13   19
  4     5    7   73   97
  5    11  241   19   37  601
  6   103  131 2161   67  277  149
  ...

Cf. A362777, A362778.

Cf. A002583 (1st column).

T(n,k) = A006530(A362777(n,k))

A070314 a(n) = P(n!+1)-P(2^n+1) where P(x) is the largest prime factor in x.

Original entry on oeis.org

0, -1, -2, 4, -12, 0, 90, 28, 404, 250, 329850, 39916118, 2834088, 75021616, 3790360374, 46271010, 993974, 956666, 123610842, 1713311273189068, 117876621366, 2703875810364, 93799610095767534, 148139754734068388, 765041185860961083618, 38681321803817920155550

Offset: 0

Benoit Cloitre, May 12 2002

sign

Is it always true that a(n) > 0 for n > 5? More generally, if m is an integer > 2, is there always an integer f(m) such that P(n!+1) > P(m^n+1) for n > f(m) (it seems that f(2) = 5, f(3) = 7, f(4) = 17, ...).

Amiram Eldar, Table of n, a(n) for n = 0..139

Cf. A002583, A002587, A006530.

gpf[n_] := FactorInteger[n][[-1,1]]; a[n_] := gpf[n!+1] - gpf[2^n+1]; Array[a, 26, 0] (* Amiram Eldar, Apr 23 2025 *)

a(n) = A002583(n) - A002587(n). - Amiram Eldar, Apr 23 2025

Offset changed to 0 and a(0) prepended by Amiram Eldar, Apr 23 2025

cp's OEIS Frontend

A038507 a(n) = n! + 1.

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A002582 Largest prime factor of n! - 1.

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A051301 Smallest prime factor of n!+1.

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A104367 Greatest prime factor of A104365(n) = A104350(n) + 1.

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A064144 a(n) is the number of divisors of n!+1.

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A164314 Largest prime factor of n^2 - 2.

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