A064237 -id:A064237 - cp's OEIS frontend

A066856 a(n) = omega(n!+1), where omega is the number of distinct primes dividing n, A001221.

1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 5, 3, 6, 2, 2, 3, 3, 3, 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

Robert G. Wilson v, Jan 21 2002

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 09 2003

Amiram Eldar, 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.
Paul Leyland, Factors of n!+1 [Typo in URL corrected by R. J. Mathar, Nov 21 2008]
Hisanori Mishima, Appendix 1. Factorization results for n!+1
Hisanori Mishima, Bernoulli numbers (n = 2 to 114)

Cf. A054990 (bigomega(n!+1)), A002981 (n!+1 is prime), A064237 (n!+1 divisible by a square), A084846 (mu(n!+1)).

[#PrimeDivisors(Factorial(n) + 1): n in [1..55]]; // Vincenzo Librandi, Oct 11 2018

Table[ Length[ FactorInteger[ n! + 1]], {n, 1, 15}]
PrimeNu[Range[50]! + 1] (* Paolo Xausa, Feb 07 2025 *)

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

More terms from Rick L. Shepherd, Jun 09 2003

A084846 mu(n!+1), where mu is the Moebius function (A008683).

Original entry on oeis.org

-1, -1, -1, -1, 0, 0, 1, 0, 1, -1, 1, -1, 0, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, 0, 1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, -1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, -1, -1, -1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1

Offset: 0

Rick L. Shepherd, Jun 10 2003

			a(6)=1 because 6!+1 = 721 = 7 * 103, the product of two different primes and thus mu(6!+1) = (-1)^2 = 1.

Amiram Eldar, Table of n, a(n) for n = 0..139
Paul Leyland, Factors of n!+1.
Markus Tervooren et al., Factorization of n!+1, FactorDB.

Cf. A008683 (mu(n)), A054990 (bigomega(n!+1)), A066856 (omega(n!+1)), A064237 (n!+1 divisible by a square), A002981 (n!+1 is prime).

[MoebiusMu(Factorial(n)+1) : n in [1..45]];

MoebiusMu[Range[0, 50]! + 1] (* Paolo Xausa, Feb 07 2025 *)

PARI
```
for(n=0,45,print1(moebius(n!+1),","))
```

If n is in A064237, then a(n) = 0. Otherwise a(n) = (-1)^A054990(n) = (-1)^A066856(n). - Max Alekseyev, Oct 08 2019

a(112) corrected, a(113)-a(114) added by Max Alekseyev, May 28 2015

a(106)-a(107) corrected by Amiram Eldar, Oct 03 2019

A092984 a(n) = the least k >= 1 such that n! + k is squarefree.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amarnath Murthy, Mar 28 2004

nonn

Conjecture: There exists a finite k such that a(n) < k for all n. Subsidiary sequence: Index of the first occurrence of n in this sequence. In case the conjecture is true, this sequence would be finite.

If a(n) = 2 ==> n!+1 is divisible by a square (sequence A064237). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 29 2004

			a(5) = 2 = 122 - 5! = 122 - 120 (as 121 = 11^2 is not squarefree).

Cf. A000142, A064237, A092983.

Table[SelectFirst[Range@ 10, SquareFreeQ[n! + #] &], {n, 45}] (* Michael De Vlieger, Aug 23 2017 *)
Table[Module[{k=1,c=n!},While[!SquareFreeQ[c+k],k++];k],{n,110}] (* Harvey P. Dale, Jul 14 2025 *)

a(n)=for(i=1,n!,if(issquarefree(n!+i),return(i)))

A092984(n) = { my(k=1); while(!issquarefree(n!+k), k++); k; }; \\ Antti Karttunen, Aug 22 2017

a(n) = A092983(n) - n!.

More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 29 2004
More terms from David Wasserman, Sep 27 2006
Typo in description corrected by Antti Karttunen, Aug 22 2017

A115091 Primes p such that p^2 divides m!+1 for some integer m

Original entry on oeis.org

5, 11, 13, 47, 71, 563, 613

Offset: 1

T. D. Noe, Mar 01 2006

By Wilson's theorem, we know that there is an m=p-1 such that p divides m!+1. Sequence A115092 gives the number of m for each prime. Occasionally p^2 also divides m!+1. These primes seem to be only slightly more plentiful than Wilson primes (A007540). No other primes < 10^6.
There is no prime p < 10^8 such that p^2 divides m!+1 for some m <= 1200. [From F. Brunault (brunault(AT)gmail.com), Nov 23 2008]
For a(n), m = p-A259230(n). - Felix Fröhlich, Jan 24 2016

R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section A2.

Cf. A064237 (n!+1 is divisible by a square), A259230.

nn=1000; lst={}; Do[p=Prime[i]; p2=p^2; f=1; m=1; While[mMichael De Vlieger, Jan 24 2016, Version 10 *)

forprime(p=1, , for(k=1, p-1, if(Mod((p-k)!, p^2)==-1, print1(p, ", "); break({1})))) \\ Felix Fröhlich, Jan 24 2016

A152219 Numbers k such that k! - 1 is divisible by a square greater than one.

Original entry on oeis.org

9, 15, 105, 112, 609, 4929

Offset: 1

Artur Jasinski, Nov 29 2008

Primes p such that p^2 divides m!-1 for some integer m

A152220.

Cf. A064237.

aa = {}; Do[If[(Sqrt[n! - 1] /. Sqrt[_] -> 1) > 1, Print[n]; AppendTo[aa, n]], {n, 1, 1000}]; aa
(* alternate program *)
nfdsQ[n_]:=AnyTrue[Rest[Divisors[n!-1]],IntegerQ[Sqrt[#]]&]; Select[Range[ 2,610],nfdsQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 24 2020 *)

a(6) from Artur Jasinski, Nov 30 2008

Definition clarified by Harvey P. Dale, May 24 2020

A152220 Primes p such that p^2 divides m!-1 for some integer m < p.

Original entry on oeis.org

11, 31, 107, 571, 971, 4931

Offset: 1

Artur Jasinski, Nov 29 2008

For numbers k such that k! - 1 is divisible by a square see A152219.

a(7) > 60000, if it exists. - Amiram Eldar, Oct 23 2024

Cf. A064237, A152219, A152231, A152232.

aa = {}; Do[If[(Sqrt[n! - 1] /. Sqrt[] -> 1) > 1, Print[(Sqrt[n! - 1] /. Sqrt[] -> 1)]; AppendTo[aa, (Sqrt[n! - 1] /. Sqrt[_] -> 1)]], {n, 1, 1000}]; aa
q[p_] := Module[{m = 2}, While[m < p && ! Divisible[m! - 1, p^2], m++]; Divisible[m! - 1, p^2]]; Select[Prime[Range[660]], q] (* Amiram Eldar, Oct 23 2024 *)

is(p) = if(isprime(p), my(m = 2); while(m < p && (m! - 1) % (p^2), m++); !((m! - 1) % (p^2)), 0); \\ Amiram Eldar, Oct 23 2024

a(6) from Artur Jasinski, Nov 30 2008

cp's OEIS Frontend

A066856 a(n) = omega(n!+1), where omega is the number of distinct primes dividing n, A001221.

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A084846 mu(n!+1), where mu is the Moebius function (A008683).

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A092984 a(n) = the least k >= 1 such that n! + k is squarefree.

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A115091 Primes p such that p^2 divides m!+1 for some integer m

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Crossrefs

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Mathematica

PARI

A152219 Numbers k such that k! - 1 is divisible by a square greater than one.

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Programs

Mathematica

Extensions

A152220 Primes p such that p^2 divides m!-1 for some integer m < p.

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Programs

Mathematica

PARI

Extensions