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A006794 Primorial -1 primes: primes p such that -1 + product of primes up to p is prime.
(Formerly M2474)

%I M2474

%S 3,5,11,13,41,89,317,337,991,1873,2053,2377,4093,4297,4583,6569,13033,

%T 15877,843301,1098133

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

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

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

%D H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203.

%D R. K. Guy, Unsolved Problems in Number Theory, Section A2.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H C. K. Caldwell, <a href="http://primes.utm.edu/primes/search.php?Description=%5E[[:digit:]]{1,}%23-1">Prime Pages: Database Search</a>

%H C. K. Caldwell, <a href="http://primes.utm.edu/glossary/page.php?sort=PrimorialPrime">Primorial Primes</a>

%H C. K. Caldwell, <a href="http://dx.doi.org/10.1090/S0025-5718-1995-1284663-3">On the primality of n! +- 1 and 2*3*5*...*p +- 1</a>, Math. Comput. 64, 889-890, 1995.

%H C. K. Caldwell and Y. Gallot, <a href="http://dx.doi.org/10.1090/S0025-5718-01-01315-1">On the primality of n!+-1 and 2*3*5*...*p+-1</a>, Math. Comp., 71 (2001), 441-448.

%H H. Dubner, <a href="/A006794/a006794.pdf">Factorial and primorial primes</a>, J. Rec. Math., 19 (No. 3, 1987), 197-203. (Annotated scanned copy)

%H Des MacHale, <a href="http://dx.doi.org/10.2307/3621650">Infinitely many proofs that there are infinitely many primes</a>, Math. Gazette, 97 (No. 540, 2013), 495-498.

%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - N. J. A. Sloane, Jun 13 2012

%H PrimeGrid, <a href="http://prpnet.primegrid.com:12008/user_primes.html">Primorial Prime Search - Primes by User</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimorialPrime.html">Primorial Prime</a>

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

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

%t primorial[p_] := Product[Prime[k], {k, 1, PrimePi[p]}]; Select[Prime[Range[1900]], PrimeQ[primorial[#] - 1] &] (* _Jean-Fran├žois Alcover_, Mar 16 2011 *)

%t Transpose[With[{pr=Prime[Range[2000]]},Select[Thread[{Rest[FoldList[ Times,1,pr]], pr}], PrimeQ[ First[#]-1]&]]][[2]] (* _Harvey P. Dale_, Jun 21 2011 *)

%t With[{p = Prime[Range[200]]}, p[[Flatten[Position[Rest[FoldList[Times, 1, p]] - 1, _?PrimeQ]]]]] (* _Eric W. Weisstein_, Nov 03 2015 *)

%o (PARI) is(n)=isprime(n) && ispseudoprime(prod(i=1,primepi(n),prime(i))-1) \\ _Charles R Greathouse IV_, Apr 29 2015

%o (Python)

%o from sympy import nextprime, isprime

%o A006794_list, p, q = [], 2, 2

%o while p < 10**5:

%o if isprime(q-1):

%o A006794_list.append(p)

%o p = nextprime(p)

%o q *= p # _Chai Wah Wu_, Apr 03 2021

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

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

%K nonn,hard,more,nice,changed

%O 1,1

%A _N. J. A. Sloane_

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

%E 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).

%E a(19)-a(20) from Eric W. Weisstein, Dec 08 2015 (Mark Rodenkirch confirms based on saved log files that all p < 700,000 have been tested)

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Last modified April 14 05:00 EDT 2021. Contains 342941 sequences. (Running on oeis4.)