A140294 -id:A140294 - cp's OEIS frontend

A049420 Composite numbers k such that k!/k# + 1 is prime, where k# = primorial numbers A034386.

4, 8, 14, 20, 26, 34, 56, 104, 153, 182, 194, 217, 230, 280, 462, 529, 1445, 2515, 3692, 6187, 6851, 13917, 17258, 48934, 83515, 96835

Offset: 1

Paul Jobling (paul.jobling(AT)whitecross.com)

Note that k!/k# is known as k compositorial.
Subset of A140294. Prime numbers are excluded since k!/k# = (k-1)!/(k-1)# when k is prime. - Giovanni Resta, Mar 28 2013
a(23) > 14000. - Giovanni Resta, Apr 02 2013
a(25) > 50000. - Roger Karpin, Jul 07 2015
The prime associated with a(26) was discovered by Serge Batalov in 2015. All k up to 10^5 were resolved by PrimeGrid administrator "Stream" (Roman Trunov) who found a(25) and the position of a(26). - Jeppe Stig Nielsen, Jul 13 2025

Chris Caldwell, Compositorial

Cf. A034386, A140293, A140294, A140315, A049421, A053982.

Primorial[n_] := Product[Prime[i], {i, 1, PrimePi[n]}];
Select[Range[2,
1000], ! PrimeQ[#] && PrimeQ[(#! / Primorial[#]) + 1] &] (* Robert Price, Oct 11 2019 *)

a(20) from Giovanni Resta, Mar 28 2013
a(21)-a(22) from Giovanni Resta, Apr 02 2013
a(23) from Roger Karpin, Nov 28 2014
a(24) from Roger Karpin, Jul 07 2015
a(25)-a(26) communicated by Jeppe Stig Nielsen, Jul 13 2025

A140293 Numbers k such that k!/k#-1 is prime, where k# is the primorial function (A034386).

Original entry on oeis.org

4, 5, 6, 7, 8, 16, 17, 21, 34, 39, 45, 50, 72, 73, 76, 133, 164, 202, 216, 221, 280, 281, 496, 605, 2532, 2967, 3337, 8711, 10977, 13724, 15250, 18160, 20943, 33684, 41400

Offset: 1

Cino Hilliard, May 25 2008

a(31) > 14000. - Giovanni Resta, Apr 02 2013
a(36) > 50000. - Roger Karpin, Jul 07 2015
If k is a prime and k is a member, then k-1 is also a member, and k!/k# - 1 is the same as (k-1)!/(k-1)# - 1. See A049421. - Jeppe Stig Nielsen, Aug 12 2024

			7!/7# = 5040/210 = 24. 24 - 1 = 23, which is prime.

Chris Caldwell, Compositorial
Chris Caldwell, Compositorial list search

Cf. A140294, A140315, A049420, A049421, A049614, A057017.

Select[Range[16], PrimeQ[#!/(Times@@Prime[Range[PrimePi[#]]]) - 1] &] (* Alonso del Arte, Nov 28 2014 *)

g(n) = for(x=4,n,y=x!/primorial(x)-1;z=nextprime(y+1); if(ispseudoprime(y),print1(x",")))

n such that n!/n# - 1 is prime, where n# is the primorial function n# = product(i = 1 .. pi(n), prime(i)), where pi(n) is the prime counting function.

a(18)-a(27) from Giovanni Resta, Mar 28 2013
a(28)-a(30) from Giovanni Resta, Apr 02 2013
a(31) from Roger Karpin, Nov 28 2014
a(32)-a(33) from Daniel Heuer, ca Aug 2000
a(34)-a(35) from Serge Batalov, Feb 09 2015

A049421 Composite numbers k such that k!/k# - 1 is prime, where k# = primorial numbers A034386.

Original entry on oeis.org

4, 6, 8, 16, 21, 34, 39, 45, 50, 72, 76, 133, 164, 202, 216, 221, 280, 496, 605, 2532, 2967, 3337, 8711, 10977, 13724, 15250, 18160, 20943, 33684, 41400

Offset: 1

Paul Jobling (paul.jobling(AT)whitecross.com)

k!/k# is known as n compositorial.
Subset of A140293. Prime numbers are excluded since k!/k# = (k-1)!/(k-1)# when k is prime. - Giovanni Resta, Mar 28 2013
a(31) > 50000. - Roger Karpin, Jul 08 2015

Chris Caldwell, Compositorial

Cf. A034386, A140293, A140294, A140315, A049420, A057017.

Primorial[n_] := Product[Prime[i], {i, 1, PrimePi[n]}];
Select[Range[2,
1000], ! PrimeQ[#] && PrimeQ[(#! / Primorial[#]) - 1] &] (* Robert Price, Oct 11 2019 *)

More terms from Robert G. Wilson v, Jun 21 2001
a(23)-a(25) from Giovanni Resta, Apr 02 2013
a(26) from Roger Karpin, Nov 29 2014
a(27)-a(28) from Daniel Heuer, ca Aug 2000
a(29)-a(30) from Serge Batalov, Feb 09 2015

A140315 Numbers k such that k!/k#-1 and k!/k#+1 are a twin prime pair.

Original entry on oeis.org

4, 5, 8, 34, 280, 281

Offset: 1

Cino Hilliard, May 25 2008

4,5 and 280,281 result in the same respective twin prime pairs. Using gmp, testing n < 4000, the last 3-prp found was the 8897 digit 3-prp, 3337!/3337#-1.

			8!/8#-1 = 191, 8!/8#-1 = 193. 191 and 193 form a twin prime pair.

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 281.

Cf. A001097, A034386, A049614, A140293, A140294.

Primorial[n_] := Product[Prime[i], {i, 1, PrimePi[n]}];
Select[Range[
  1000], (p = (#! / Primorial[#]);
PrimeQ[p + 1] && PrimeQ[p - 1]) &] (* Robert Price, Oct 11 2019 *)

g(n) = for(x=1,n,y=x!/primorial(x)-1;z=nextprime(y+1); if(ispseudoprime(y)&&z-y==2,print1(x", ")))
primorial(n) = { local(p1,x); if(n==0||n==1,return(1)); p1=1; forprime(x=2,n,p1*=x); return(p1) }

n# is the primorial function A034386(n).

A140293 INTERSECT A140294. - R. J. Mathar, Feb 27 2012

Offset corrected by Amiram Eldar, Jul 18 2025

A053982 Numbers k such that 1 + product of first k composite numbers is prime.

Original entry on oeis.org

1, 3, 7, 11, 16, 22, 39, 76, 116, 139, 149, 169, 179, 220, 372, 429, 1216, 2146, 3176, 5382, 5969, 12271, 15271, 43903

Offset: 1

G. L. Honaker, Jr., Apr 02 2000

Cf. A002808, A000040, A049420, A049650, A057017, A140294.

Composite[n_Integer] := (k = n + PrimePi[n] + 1; While[k - PrimePi[k] - 1 != n, k++ ]; k); Do[ If[ PrimeQ[ Product[ Composite[k], {k, 1, n} ] + 1], Print[ n ] ], {n, 1, 430} ]
Position[FoldList[Times,Select[Range[1500],CompositeQ]],?(PrimeQ[#+1]&)]//Flatten (* _Harvey P. Dale, Dec 20 2022 *)

lista(kmax) = {my(m = 1, k = 0); forcomposite(c = 1, , k++; if(k > kmax, break); m *= c; if(isprime(m+1), print1(k, ", ")));} \\ Amiram Eldar, Jun 03 2024

More terms from Jeppe Stig Nielsen, Apr 16 2000 (terms from 76 on correspond to probable primes)
a(16)-a(17) from Robert G. Wilson v, Apr 20 2001
Edited by T. D. Noe, Oct 30 2008
a(18)-a(19) from Amiram Eldar, Jun 03 2024
a(20)-a(21) from Michael S. Branicky, Jun 04 2024
More terms via A049420 from Jeppe Stig Nielsen, Aug 12 2024

A222255 Primes of the form n!/n# + 1, where n#=A034386(n) (primorial), listed with repetition.

Original entry on oeis.org

2, 2, 2, 2, 5, 5, 193, 2903041, 250822656001, 1807729046323200001, 1472038679443987759104000001, 21817028147643299474152432146548259236610048000000000001

Offset: 1

M. F. Hasler, Mar 27 2013

nonn

For each n >= 0, if n!/n#+1 is prime, then this prime is listed here: This explains the repetitions.

The next term is already 126 digits long.

for(n=0,99,ispseudoprime(a=n!/prod(k=1,primepi(n),prime(k))+1)&print1(a","))

a(n) = A049614(A140294(n))+1, where A049614 = A000142/A034386.

cp's OEIS Frontend

A049420 Composite numbers k such that k!/k# + 1 is prime, where k# = primorial numbers A034386.

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A140293 Numbers k such that k!/k#-1 is prime, where k# is the primorial function (A034386).

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A049421 Composite numbers k such that k!/k# - 1 is prime, where k# = primorial numbers A034386.

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A140315 Numbers k such that k!/k#-1 and k!/k#+1 are a twin prime pair.

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A053982 Numbers k such that 1 + product of first k composite numbers is prime.

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A222255 Primes of the form n!/n# + 1, where n#=A034386(n) (primorial), listed with repetition.

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