A084846 -id:A084846 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A063684 Numbers k such that m(k!) = 2, where m(k) = mu(k) + mu(k+1) + mu(k+2).

Original entry on oeis.org

8, 13, 14, 18, 19, 20, 25, 36, 38, 43, 48, 51, 52, 54, 60, 71, 74, 75, 78, 80, 87, 91, 92, 105, 108, 110, 112, 114

Offset: 1

Jason Earls, Aug 22 2001

Equivalently, k such that m(k!) = 2, where m(k) = mu(k+1) + mu(k+2), as mu(k!)=0 for all k >= 4 (because 4=2^2 divides k!). - Rick L. Shepherd, Aug 20 2003

127 belongs to the sequence. - Serge Batalov, Feb 17 2011

			8 is a term: 8! = 40320; mu(40320) = 0, mu(40321) = 1, mu(40322) = 1, 0+1+1 = 2.
98 is not a term because 98! + 2 = 2 * 31003012014959 * 114951592532951 * 2015644865638913835753087050212028452990938458387 * P78 has an odd number of factors. - _Sean A. Irvine_, Feb 03 2010

Dario A. Alpern, Factorization using the Elliptic Curve Method.
Paul Leyland, Factors of n!+1 (updated 1 Oct 2006).

Cf. A063838, A008683.

Cf. A084846 (mu(n!+1)).

m(n) = moebius(n)+moebius(n+1)+moebius(n+2); for(n=1,10^4, if(m(n!)==2,print(n)))

More terms from Rick L. Shepherd, Aug 20 2003
Two more terms from Sean A. Irvine, Feb 03 2010, Feb 08 2010
Two new terms, 105 and 108, from Daniel M. Jensen, Feb 19 2011, Mar 02 2011
Two more terms, 110 and 112, from Serge Batalov, Mar 04-05 2011
One more term, 114, from Sean A. Irvine, May 25 2015

cp's OEIS Frontend

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

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

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Magma

Mathematica

PARI

Extensions

A063684 Numbers k such that m(k!) = 2, where m(k) = mu(k) + mu(k+1) + mu(k+2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions