A053982 -id:A053982 - 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

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

Original entry on oeis.org

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

A057017 Numbers k such that the product of the first k composite numbers minus 1 is a prime.

Original entry on oeis.org

1, 2, 3, 9, 12, 22, 26, 30, 34, 51, 54, 100, 125, 155, 168, 173, 220, 401, 494, 2161, 2539, 2866, 7625, 9644, 12099, 13470, 16078, 18587, 30075, 37067

Offset: 1

Robert G. Wilson v, Apr 21 2001

nonn

			a(3) = 3 because 4*6*8-1 = 191 which is prime.

Cf. A049421, A053982, A140293.

Composite[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1 != n, k++ ]; k); Do[ m = n; If[ PrimeQ[ Product[ Composite[ k ], {k, 1, n} ] - 1 ], Print[ n ] ], {n, 1, 1980} ]

More terms via A049421 from Jeppe Stig Nielsen, Aug 12 2024

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

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A057017 Numbers k such that the product of the first k composite numbers minus 1 is a prime.

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