A049420 -id:A049420 - cp's OEIS frontend

A140294 Numbers k such that k!/k# + 1 is prime, where k# is the primorial function (A034386).

0, 1, 2, 3, 4, 5, 8, 14, 20, 26, 34, 56, 104, 153, 182, 194, 217, 230, 280, 281, 462, 463, 529, 1445, 2515, 3692, 6187, 6851, 13917, 17258, 48934, 83515, 96835

Offset: 1

Cino Hilliard, May 25 2008

nonn

96835 is a term of the sequence, but its rank is not currently known. - Serge Batalov, Feb 06 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 A049420. - Jeppe Stig Nielsen, Aug 12 2024
All k up to 10^5 were resolved by PrimeGrid administrator "Stream" (Roman Trunov) who found a(32) and found the position of term mentioned by Batalov above (it is a(33)). - Jeppe Stig Nielsen, Jul 13 2025

			8!/8# + 1 = 40320/210 + 1 = 193, a prime.

Chris Caldwell, Compositorial

Cf. A034386, A140293, A140315, A049420, A049421, A053982.

A140294 := proc(n) local L, p, s, i; L := 1;
for p in select(isprime, [$2..iquo(n,2)]) do
    s := add(i,i=convert(n,base,p)); L := L*p^((n-s)/(p-1)-1) od;
`if`(isprime(L+1), n, NULL) end:
seq(A140294(i), i=0..104); # Peter Luschny, Mar 27 2013

Primorial[p_] := Times @@ Prime[Range[PrimePi[p]]]; Select[Range[0,194], PrimeQ[#!/Primorial[#] + 1] &] (* T. D. Noe, Mar 27 2013 *)

is(n)=ispseudoprime(n!/prod(i=1,primepi(n),prime(i))+1) \\ Charles R Greathouse IV, Mar 27 2013

PFGW
```
ABC2 $a!/$a#+1
a: from 1 to 3000
```

a(17)-a(25) from Charles R Greathouse IV, Mar 27 2013
a(26)-a(27) from Giovanni Resta, Mar 28 2013
a(28) from Charles R Greathouse IV, Mar 28 2013
a(29) from Giovanni Resta, Apr 02 2013
a(30) from Roger Karpin, Nov 29 2014
a(31) from Roger Karpin, Jun 08 2015
a(32)-a(33) 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

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

A108894 Numbers k such that (k!/k#) * 2^k + 1 is prime, where n# = primorial numbers (A034386).

Original entry on oeis.org

0, 1, 2, 11, 17, 25, 38, 53, 107, 245, 255, 367, 719, 1077, 2189, 2853, 3236, 3511, 3633, 4531, 4858, 5422, 7787, 8319

Offset: 1

Jason Earls, Jul 15 2005

n!/n# is known as n compositorial. All values have been proved prime. No more terms up to 6100. Primality proof for the largest, which has 17219 digits: PFGW Version 1.2.0 for Windows [FFT v23.8] Primality testing (5422!/5422#)*(2^5422)+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 2719 Calling Brillhart-Lehmer-Selfridge with factored part 36.34% (5422!/5422#)*(2^5422)+1 is prime! (66.5095s+0.0129s)

Cf. A049420, A091421.

f[n_] := n!/Fold[Times, 1, Prime[ Range[ PrimePi[ n]]]]*2^n + 1; Do[ If[ PrimeQ[ f[n]], Print[n]], {n, 0, 1100}] (* Robert G. Wilson v, Jul 18 2005 *)

a(23)-a(24) from Michael S. Branicky, Oct 01 2024

A085274 Composite k such that (k!/k#) + 1 is a semiprime, where k# = primorial numbers A034386.

Original entry on oeis.org

6, 10, 21, 22, 24, 25, 27, 30, 39, 48, 52, 57, 65, 87, 94, 110, 114, 124, 156, 161

Offset: 1

Jason Earls, Aug 12 2003

nonn

n!/n# is called n compositorial. The actual sequence is (6,7),(10,11),21,(22,23),24,25,27,(30,31),39,48, (52,53),57,65,... where the values in parenthesis yield the same semiprime.

That is, since p!/p# = (p-1)!/(p-1)#, primes never appear in this sequence. - Sean A. Irvine, Jun 30 2020

			25!/25# + 1 is a product of two primes: 69528040243200001 = 2594807 * 26795071943.

Hisanori Mishima, Compositorial + 1 (n = 4 to 150).

Cf. A001358, A049420, A049614.

More terms from Robert G. Wilson v, Aug 15 2003

Offset corrected and a(19) and a(20) from Sean A. Irvine, Jun 30 2020

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A140294 Numbers k such that k!/k# + 1 is prime, where k# is the primorial function (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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A053982 Numbers k such that 1 + product of first k composite numbers is prime.

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A108894 Numbers k such that (k!/k#) * 2^k + 1 is prime, where n# = primorial numbers (A034386).

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A085274 Composite k such that (k!/k#) + 1 is a semiprime, where k# = primorial numbers A034386.

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