A038507 -id:A038507 - cp's OEIS frontend

A002582 Largest prime factor of n! - 1.

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

A085692 Brocard's problem: squares which can be written as n!+1 for some n.

Original entry on oeis.org

25, 121, 5041

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 18 2003

Next term, if it exists, is greater than 10^850. - Sascha Kurz, Sep 22 2003
No more terms < 10^20000. - David Wasserman, Feb 08 2005
The problem of whether there are any other terms in this sequence, Brocard's problem, has been unsolved since 1876. The known calculations give a(4) > (10^9)! = factorial(10^9). - Stefan Steinerberger, Mar 19 2006
I wrote a similar program sieving against the 40 smallest primes larger than 4*10^9 and can report that a(4) > factorial(4*10^9+1). In other words, it's now known that the only n <= 4*10^9 for which n!+1 is a square are 4, 5 and 7. C source code available on request. - Tim Peters (tim.one(AT)comcast.net), Jul 02 2006
Robert Matson claims to have verified that 4, 5, and 7 are the only values of n <= 10^12 for which n!+1 is a square. This implies that the next term, if it exists, is greater than (10^12+1)! ~ 1.4*10^11565705518115. - David Radcliffe, Oct 28 2019

			   5^2 =   25 = 4! + 1;
  11^2 =  121 = 5! + 1;
  71^2 = 5041 = 7! + 1.

R. Guy, "Unsolved Problems in Number Theory", 3rd edition, D25
Clifford A. Pickover, A Passion for Mathematics (2005) at 69, 306.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 19.

Bruce C. Berndt and William F. Galway, On the Brocard-Ramanujan Diophantine Equation n! + 1 = m^2, The Ramanujan Journal, March 2000, Volume 4, Issue 1, pp 41-42.
Robert D. Matson, Brocard's Problem 4th Solution Search Utilizing Quadratic Residues, Unsolved Problems.
Wikipedia, Brocard's problem.

A085692, A146968, A216071 are all essentially the same sequence. - N. J. A. Sloane, Sep 01 2012

Select[Range[0,100]!+1,IntegerQ[Sqrt[#]] &] (* Stefano Spezia, Jul 02 2025 *)

A085692=select( issquare, vector(99,n,n!+1)) \\ M. F. Hasler, Nov 20 2018

a(n) = A216071(n)^2 = A146968(n)!+1 = A038507(A146968(n)). - M. F. Hasler, Nov 20 2018

A002585 Largest prime factor of 1 + (product of first n primes).

Original entry on oeis.org

3, 7, 31, 211, 2311, 509, 277, 27953, 703763, 34231, 200560490131, 676421, 11072701, 78339888213593, 13808181181, 18564761860301, 19026377261, 525956867082542470777, 143581524529603, 2892214489673, 16156160491570418147806951, 96888414202798247, 1004988035964897329167431269

Offset: 1

N. J. A. Sloane

Based on Euclid's proof that there are infinitely many primes.

The products of the first primes are called primorial numbers. - Franklin T. Adams-Watters, Jun 12 2014

M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
M. Kraitchik, Introduction à la Théorie des Nombres. Gauthier-Villars, Paris, 1952, p. 2.
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).

Table of n, a(n) for n = 1..98
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96.
S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96. [Annotated scanned copy; also letter from N. J. A. Sloane to John Selfridge]
R. 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], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
John Selfridge, Marvin Wunderlich, Robert Morris, N. J. A. Sloane, Correspondence, 1975
Eric Weisstein's World of Mathematics, Euclid Number.
R. G. Wilson v, Explicit factorizations

Cf. A002110, A002584, A006530, A006862, A051342.

FactorInteger[#][[-1,1]]&/@Rest[FoldList[Times,1,Prime[Range[30]]]+1] (* Harvey P. Dale, Apr 10 2012 *)

a(n)=my(f=factor(prod(i=1,n,prime(i))+1)[,1]); f[#f] \\ Charles R Greathouse IV, Feb 07 2017

a(n) = A006530(A006862(n)). - Amiram Eldar, Feb 13 2020

More terms from Labos Elemer, May 02 2000
More terms from Robert G. Wilson v, Mar 24 2001
Terms up to a(81) in b-file added by Sean A. Irvine, Apr 19 2014
Terms a(82)-a(87) in b-file added by Amiram Eldar, Feb 13 2020
Terms a(88)-a(98) in b-file added by Max Alekseyev, Aug 26 2021

A005095 a(n) = n! + n.

Original entry on oeis.org

1, 2, 4, 9, 28, 125, 726, 5047, 40328, 362889, 3628810, 39916811, 479001612, 6227020813, 87178291214, 1307674368015, 20922789888016, 355687428096017, 6402373705728018, 121645100408832019, 2432902008176640020, 51090942171709440021

Offset: 0

N. J. A. Sloane

Every infinite, increasing, integer arithmetic progression meets this sequence infinitely often. - John Abbott (abbott(AT)dima.unige.it), Mar 06 2003
Sum(A010051(k): A038507(n) < k <= a(n)) = 0. - Reinhard Zumkeller, Jul 10 2009
Largest k such that (k!-n!)/(k-n) is an integer. - Derek Orr, Apr 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, Card Tricks and Information, arXiv:2405.21007 [math.HO], 2024. See p. 5.
Index entries for sequences related to factorial numbers

Cf. A135723.

Cf. A090786. - Reinhard Zumkeller, Jul 10 2009

[Factorial(n) + n: n in [0..20]]; // Vincenzo Librandi, Jun 08 2013

lst={};Do[AppendTo[lst, n!+n], {n, 3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 20 2008 *)
Table[n! + n, {n, 0, 20}] (* Vincenzo Librandi, Jun 08 2013 *)

A005095(n):= n!+n$
makelist(A005095(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

E.g.f.: x*exp(x) + 1/(1-x). - Len Smiley, Dec 05 2001
Row sums of triangle A135723. - Gary W. Adamson, Nov 25 2007
(n-1)*(n-3)*a(n) -n*(n^2-3*n+1)*a(n-1) +n*(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Oct 30 2015
a(n) +(-n-3)*a(n-1) +3*(n)*a(n-2) +(-3*n+5)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Oct 30 2015

A054989 Number of prime divisors of -1 + (product of first n primes).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 3, 2, 2, 4, 1, 2, 3, 3, 2, 3, 3, 2, 2, 4, 3, 1, 2, 2, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 4, 4, 2, 3, 3, 2, 4, 3, 4, 2, 4, 4, 7, 4, 3, 3, 4, 4, 3, 3, 1, 3, 1, 4, 3, 5, 5, 4, 4, 6, 5, 5, 3, 4, 3, 4, 4, 3, 4, 2, 3, 4

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

			a(4)=2 because 2*3*5*7 - 1 = 209 = 11*19

Amiram Eldar, Table of n, a(n) for n = 1..99
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A054988, A054990, A054991, A054992, A057588.

a[q_] := Module[{x, n}, x=FactorInteger[Product[Table[Prime[i], {i, q}][[j]], {j, q}]-1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
PrimeOmega[#] & /@ (FoldList[Times, Prime[Range[81]]] - 1) (* Harvey P. Dale, Mar 11 2017 *)

More terms from Robert G. Wilson v, Mar 24 2001
a(42)-a(81) from Charles R Greathouse IV, May 07 2011
a(82)-a(87) from Amiram Eldar, Oct 03 2019

A054990 Number of prime divisors of n! + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 3, 2, 2, 3, 5, 3, 6, 2, 2, 3, 3, 4, 2, 2, 2, 1, 2, 3, 5, 4, 4, 5, 2, 5, 6, 1, 2, 4, 7, 1, 3, 4, 3, 3, 3, 4, 2, 5, 5, 6, 4, 4, 2, 2, 4, 3, 4, 2, 4, 4, 3, 5, 3, 4, 5, 4, 5, 6, 5, 2, 7, 1, 4, 2, 3, 1, 6, 3, 4, 7, 3, 3, 3, 5, 5, 4, 3, 8, 3, 6, 2, 4, 3, 4, 5, 6, 6, 5, 5, 4, 5

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

The smallest k! with n prime factors occurs for n in A060250.

103!+1 = 27437*31084943*C153, so a(103) is unknown until this 153-digit composite is factored. a(104) = 4 and a(105) = 6. - Rick L. Shepherd, Jun 10 2003

			a(2)=2 because 4! + 1 = 25 = 5*5

Amiram Eldar, Table of n, a(n) for n = 1..139
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations
Paul Leyland, Factors of n!+1.

Cf. A000040 (prime numbers), A001359 (twin primes).
Cf. A038507, A054988, A054989, A054991, A054992.
Cf. A066856 (number of distinct prime divisors of n!+1), A084846 (mu(n!+1)).

a[q_] := Module[{x, n}, x=FactorInteger[q!+1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
A054990[n_Integer] := PrimeOmega[n! + 1]; Table[A054990[n], {n,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

PARI
```
for(n=1,64,print1(bigomega(n!+1),","))
```

More terms from Robert G. Wilson v, Mar 23 2001

More terms from Rick L. Shepherd, Jun 10 2003

A054988 Number of prime divisors of 1 + (product of first n primes), with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 2, 3, 1, 3, 3, 2, 3, 4, 4, 2, 2, 4, 2, 3, 2, 4, 3, 2, 4, 4, 3, 3, 5, 3, 6, 2, 3, 2, 5, 4, 4, 2, 6, 3, 4, 3, 5, 6, 7, 2, 6, 3, 5, 3, 4, 2, 6, 5, 4, 5, 3, 5, 5, 5, 3, 3, 5, 5, 6, 3, 4, 4, 7, 5, 3, 4, 1, 2, 5, 5, 5, 4, 5, 3, 5, 4, 6, 5, 8

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

Prime divisors are counted with multiplicity. - Harvey P. Dale, Oct 23 2020

It is an open question as to whether omega(p#+1) = bigomega(p#+1) = a(n); that is, as to whether the Euclid numbers are squarefree. Any square dividing p#+1 must exceed 2.5*10^15 (see Vardi, p. 87). - Sean A. Irvine, Oct 21 2023

			a(6)=2 because 2*3*5*7*11*13+1 = 30031 = 59 * 509.

Ilan Vardi, Computational Recreations in Mathematica, Addison-Wesley, 1991.

Max Alekseyev, Table of n, a(n) for n = 1..98
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A001222, A002585, A006862, A014545, A054989, A054990, A054991, A054992.

A054988 := proc(n)
    numtheory[bigomega](1+mul(ithprime(i),i=1..n)) ;
end proc:
seq(A054988(n),n=1..20) ; # R. J. Mathar, Mar 09 2022

a[q_] := Module[{x, n}, x=FactorInteger[Product[Table[Prime[i], {i, q}][[j]], {j, q}]+1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
PrimeOmega[#+1]&/@FoldList[Times,Prime[Range[90]]] (* Harvey P. Dale, Oct 23 2020 *)

a(n) = bigomega(1+prod(k=1, n, prime(k))); \\ Michel Marcus, Mar 07 2022

a(n) = Omega(1 + Product_{k=1..n} prime(k)). - Wesley Ivan Hurt, Mar 06 2022
a(n) = A001222(A006862(n)). - Michel Marcus, Mar 07 2022
a(n) = 1 iff n is in A014545. - Bernard Schott, Mar 07 2022

More terms from Robert G. Wilson v, Mar 24 2001
a(44)-a(81) from Charles R Greathouse IV, May 07 2011
a(82)-a(87) from Amiram Eldar, Oct 03 2019

A104365 a(n) = A104350(n) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 1261, 2521, 7561, 37801, 415801, 1247401, 16216201, 113513401, 567567001, 1135134001, 19297278001, 57891834001, 1099944846001, 5499724230001, 38498069610001, 423478765710001, 9740011611330001

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A006530, A006862, A038507, A104366, A104367, A104368, A104369, A104370, A104371, A104357.

FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 30]] + 1 (* Amiram Eldar, Apr 08 2024 *)

a(n) = (a(n-1) - 1) * A006530(n) + 1 for n>1, a(1) = 0;

A051342 Smallest prime factor of 1 + (product of first n primes).

Original entry on oeis.org

3, 7, 31, 211, 2311, 59, 19, 347, 317, 331, 200560490131, 181, 61, 167, 953, 73, 277, 223, 54730729297, 1063, 2521, 22093, 265739, 131, 2336993, 960703, 2297, 149, 334507, 5122427, 1543, 1951, 881, 678279959005528882498681487, 87549524399, 23269086799180847

Offset: 1

Labos Elemer

nonn

Based on Euclid's proof that there are infinitely many primes.

Amiram Eldar, Table of n, a(n) for n = 1..87
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. A014545, A002585.

a:= proc(n)
local N, F, i;
  N:= 1 + mul(ithprime(i),i=1..n);
  F:= select(type,map(t->t[1],ifactors(N,easy)[2]),integer);
if nops(F) >= 1 then return min(F) fi;
  min(map(t->t[1],ifactors(N)[2]))
end proc:
seq(a(n),n=1..40); # Robert Israel, Oct 19 2014

Map[FactorInteger,
   Table[Product[Prime[n], {n, 1, m}] + 1, {m, 1, 36}]][[All,
1]][[All, 1]] (* Geoffrey Critzer, Oct 19 2014 *)
FactorInteger[#][[1,1]]&/@(FoldList[Times,Prime[Range[40]]]+1) (* Harvey P. Dale, Oct 08 2021 *)

a(n) = factor(1 + prod(i=1, n, prime(i)))[1, 1]; \\ Michel Marcus, Dec 10 2013

a(n) = A020639(1+A002110(n)).

One more term from Michel Marcus, Dec 10 2013

cp's OEIS Frontend

A002582 Largest prime factor of n! - 1.

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Magma

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A051301 Smallest prime factor of n!+1.

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Maple

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A085692 Brocard's problem: squares which can be written as n!+1 for some n.

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Mathematica

PARI

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A002585 Largest prime factor of 1 + (product of first n primes).

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Mathematica

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A005095 a(n) = n! + n.

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Magma

Mathematica

Maxima

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A054989 Number of prime divisors of -1 + (product of first n primes).

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Examples

Links

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Programs

Mathematica

Extensions

A054990 Number of prime divisors of n! + 1 (counted with multiplicity).

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Author

Keywords