A066856 -id:A066856 - 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

A104368 Number of distinct prime factors of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

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

Cf. A001221, A054988, A066856, A104350, A104359, A104360, A104365, A104366, A104367, A104369, A104370, A104371, A104372.

A104350[n_] := Product[FactorInteger[k][[-1, 1]], {k, 1, n}]; PrimeNu[
Table[A104350[n] + 1, {n, 2, 50}]] (* G. C. Greubel, May 10 2017 *)

a(n) = A001221(A104365(n)).

a(51)-a(76) from Amiram Eldar, Feb 12 2020
a(77)-a(81) from Jinyuan Wang, Apr 02 2020
Terms a(82) onward from Max Alekseyev, Oct 03 2022

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

A066835 a(n) = omega((prime(n)-1)! + 1), where omega is given by A001221, primes in A000040.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jan 20 2002

			a(7) = omega((prime(7)-1)! + 1) = omega((17-1)! + 1) = omega(16! + 1) = omega(20922789888000 + 1) = omega(20922789888001) = 5, as 20922789888001 = 17 * 61 * 137 * 139 * 1059511 = prime(7)*prime(18)*prime(33)*prime(34)*prime(82801).

Hisanori Mishima, Appendix 1. Factorization results for n!+1
Eric Weisstein's World of Mathematics, Wilson's Theorem

Cf. A000040, A000142, A001221, A006093, A066856.

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

a(n) = omega((prime(n)-1)! + 1); \\ Jinyuan Wang, Apr 01 2020

a(n) = A066856(A006093(n)). - Jinyuan Wang, Apr 01 2020

More terms from Robert G. Wilson v, Jan 21 2002

a(27)-a(34) from Jinyuan Wang, Apr 01 2020

A180325 a(n) = k is the smallest number such that n is the number of distinct primes dividing k! + 1.

Original entry on oeis.org

1, 6, 9, 31, 16, 18, 40, 89

Offset: 1

Michel Lagneau, Jan 18 2011

A greedy inverse to A066856. - R. J. Mathar, Jan 21 2011

			a(6) = 18 because the 6 distinct primes dividing 18! + 1 = 6402373705728001
  are {19, 23, 29, 61, 67, 123610951}.

Cf. A038507.

with(numtheory):for n from 1 to 7 do:ind:=0:for k from 1 to 50 while(ind=0)
  do: x:=k!+1:y:=nops(factorset(x)):if y=n then ind:=1:printf(`%d, `,k):else fi:od:
  od:

cp's OEIS Frontend

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

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A104368 Number of distinct prime factors of A104365(n) = A104350(n) + 1.

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

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A066835 a(n) = omega((prime(n)-1)! + 1), where omega is given by A001221, primes in A000040.

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A180325 a(n) = k is the smallest number such that n is the number of distinct primes dividing k! + 1.

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Maple