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 A005234 Primorial plus 1 primes: primes p such that 1 + product of primes up to p is prime. (Formerly M0669) 22
 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Conjecture: if p# + 1 is a prime number, then the next prime is less than p# + exp(1)*p. - Arkadiusz Wesolowski, Feb 20 2013 Conjecture: if p# + 1 is a prime, then the next prime is less than p# + p^2. - Thomas Ordowski, Apr 07 2013 REFERENCES J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008. H. Dubner, A new primorial prime, J. Rec. Math., 21 (No. 4, 1989), 276. R. K. Guy, Unsolved Problems in Number Theory, Section A2. F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 109, 1983. Paulo Ribenboim, The New Book of Prime Number Records, p. 13. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS C. K. Caldwell, Prime Pages: Database Search C. K. Caldwell, Primorial Primes 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) H. Dubner, A new primorial prime, J. Rec. Math., 21.4 (1989), 276. (Annotated scanned copy) H. Dubner & N. J. A. Sloane, Correspondence, 1991 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:1202.3670 [math.HO], 2012. R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations Eric Weisstein's World of Mathematics, Euclid Number Eric Weisstein's World of Mathematics, Primorial Prime FORMULA a(n) = A000040(A014545(n+1)). - M. F. Hasler, May 31 2018 MAPLE N:= 5000: # to get all terms <= N Primes:= select(isprime, [\$2..N]): P:= 1: count:= 0: for n from 1 to nops(Primes) do    P:= P*Primes[n];    if isprime(P+1) then      count:= count+1; A[count]:= Primes[n]    fi od: seq(A[i], i=1..count); # Robert Israel, Nov 03 2015 MATHEMATICA (* This program is not convenient for large values of p *) p = pp = 1; Reap[While[p < 5000, p = NextPrime[p]; pp = pp*p; If[PrimeQ[1 + pp], Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 31 2012 *) With[{p = Prime[Range]}, p[[Flatten[Position[Rest[FoldList[Times, 1, p]] + 1, _?PrimeQ]]]]] (* Eric W. Weisstein, Nov 03 2015 *) PROG (PARI) is(n)=isprime(n) && ispseudoprime(prod(i=1, primepi(n), prime(i))+1) \\ Charles R Greathouse IV, Feb 20 2013 (PARI) is(n)=isprime(n) && ispseudoprime(factorback(primes([2, n]))+1) \\ M. F. Hasler, May 31 2018 (MAGMA) [p:p in PrimesUpTo(3000)|IsPrime(&*PrimesUpTo(p)+1)]; // Marius A. Burtea, Mar 25 2019 CROSSREFS Cf. A006862 (Euclid numbers). Cf. A014545 (Primorial plus 1 prime indices: n such that 1 + (Product of first n primes) is prime). Cf. A018239 (Primorial plus 1 primes). Cf. A002110, A006794, A057704. Sequence in context: A119388 A093487 A067933 * A254225 A140561 A140553 Adjacent sequences:  A005231 A005232 A005233 * A005235 A005236 A005237 KEYWORD nonn,hard,more,nice AUTHOR EXTENSIONS 42209 sent in by Chris Nash (chrisnash(AT)cwix.com). 145823 discovered and sent in by Arlin Anderson (starship1(AT)gmail.com) and Don Robinson (donald.robinson(AT)itt.com), Jun 01 2000 366439, 392113 from Eric W. Weisstein, Mar 13 2004 (based on information in A014545) STATUS approved

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Last modified June 25 07:53 EDT 2019. Contains 324347 sequences. (Running on oeis4.)