A005234 - OEIS

A005234

Primorial plus 1 primes: primes p such that 1 + product of primes up to p is prime.
(Formerly M0669)

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

(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).

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 112.

LINKS

Table of n, a(n) for n=1..25.

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.

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]

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 *)

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 A334026 A140561

Adjacent sequences: A005231 A005232 A005233 * A005235 A005236 A005237

KEYWORD

nonn,hard,more,nice

AUTHOR

N. J. A. Sloane

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

STATUS

approved