A006794 Primorial -1 primes: primes p such that -1 + product of primes up to p is prime.

3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, 3267113, 4778027, 6354977, 6533299

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

Comments

Or, p such that primorial(p) - 1 is prime.

Conjecture: if p# - 1 is a prime number, then the previous prime is greater than p# - exp(1)*p. - Arkadiusz Wesolowski, Jun 19 2016

References

• H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203.
• R. K. Guy, Unsolved Problems in Number Theory, Section A2.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 4-5.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 111.

Links

• C. K. Caldwell, Prime Pages: Database Search.
• C. K. Caldwell, Primorial Primes.
• C. K. Caldwell, On the primality of n! +- 1 and 2*3*5*...*p +- 1, Math. Comput. 64, 889-890, 1995.
• C. K. Caldwell and Y. Gallot, On the primality of n!+-1 and 2*3*5*...*p+-1, Math. Comp., 71 (2001), 441-448.
• H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203. (Annotated scanned copy)
• Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.
• R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
• PrimeGrid, Primorial Prime Search - Primes by User.
• Eric Weisstein's World of Mathematics, Primorial Prime.

Crossrefs

Cf. A057704 (Primorial - 1 prime indices: integers n such that the n-th primorial minus 1 is prime).

Cf. A057705, A002110, A005234, A014545, A018239.

Programs

Mathematica

primorial[p_] := Product[Prime[k], {k, 1, PrimePi[p]}]; Select[Prime[Range[1900]], PrimeQ[primorial[#] - 1] &] (* Jean-François Alcover, Mar 16 2011 *)
Transpose[With[{pr=Prime[Range[2000]]},Select[Thread[{Rest[FoldList[ Times,1,pr]], pr}], PrimeQ[ First[#]-1]&]]][[2]] (* Harvey P. Dale, Jun 21 2011 *)
With[{p = Prime[Range[200]]}, p[[Flatten[Position[Rest[FoldList[Times, 1, p]] - 1, ?PrimeQ]]]]] (* _Eric W. Weisstein, Nov 03 2015 *)

PARI

is(n)=isprime(n) && ispseudoprime(prod(i=1,primepi(n),prime(i))-1) \\ Charles R Greathouse IV, Apr 29 2015

Python

from sympy import nextprime, isprime
A006794_list, p, q = [], 2, 2
while p < 10**5:
    if isprime(q-1):
        A006794_list.append(p)
    p = nextprime(p)
    q *= p # Chai Wah Wu, Apr 03 2021

Formula

a(n) = A000040(A057704(n)).

a(n) = prime(A057704(n)).

Extensions

Stated incorrectly in CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 101; corrected in 2nd printing.

Corrected by Arlin Anderson (starship1(AT)gmail.com), who reports that he and Don Robinson have checked this sequence through about 63000 digits without finding another term (Jul 04 2000).

a(19)-a(20) from Eric W. Weisstein, Dec 08 2015 (Mark Rodenkirch confirms based on saved log files that all p < 700000 have been tested)

a(21) from Jeppe Stig Nielsen, Oct 19 2021

a(22)-a(24) from Jeppe Stig Nielsen, Dec 16 2024