A005234 Primes p such that 1 + product of primes up to p is prime.

2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, 4328927, 5256037, 6369619, 7351117, 9562633

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice
• changed

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

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 134.
• 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.
• 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. 112.

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 and 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.

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.

Programs

Magma

[p:p in PrimesUpTo(3000)|IsPrime(&*PrimesUpTo(p)+1)]; // Marius A. Burtea, Mar 25 2019

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[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, Feb 20 2013

PARI

is(n)=isprime(n) && ispseudoprime(factorback(primes([2,n]))+1) \\ M. F. Hasler, May 31 2018

Formula

a(n) = A000040(A014545(n+1)). - M. F. Hasler, May 31 2018

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)

a(23) from Jeppe Stig Nielsen, Aug 08 2024

a(24) from Jeppe Stig Nielsen, Sep 01 2024

a(25) from Jeppe Stig Nielsen, Sep 24 2024

a(26) from Jeppe Stig Nielsen, Nov 10 2024

a(27) from Jeppe Stig Nielsen, Aug 21 2025