A014545 Numbers k such that the k-th Euclid number A006862(k) = 1 + (Product of first k primes) is prime.

0, 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 304723, 365071, 436504, 498865, 637491

Offset: 1

Author

Eric W. Weisstein, Murray R. Bremner

Keywords

• nonn
• nice
• hard
• more
• changed

Examples

			a(1) = 0 because the (empty) product of 0 primes is 1, plus 1 yields the prime 2.
prime(4413) = 42209 and Primorial(4413) + 1 = 42209# + 1 is a 18241-digit prime.
prime(13494) = 145823 and Primorial(13494) + 1 = 145823# + 1 is a 63142-digit prime.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

Links

• C. K. Caldwell, Prime Pages: Database Search
• C. K. Caldwell, Primorial Primes.
• H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
• Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola (2018) Vol. 54, Issue 3.
• Eric Weisstein's World of Mathematics, Euclid Number
• Eric Weisstein's World of Mathematics, Primorial Prime
• Eric Weisstein's World of Mathematics, Integer Sequence Primes

Crossrefs

Cf. A005234 (values of p such that 1 + product of primes <= p is prime).

Cf. A018239 (primorial plus 1 primes).

Cf. A002110, A006862, A057704.

Programs

Maple

P:= 1:
p:= 1:
count:= 0:
for n from 1 to 1000 do
  p:= nextprime(p);
  P:= P*p;
  if isprime(P+1) then
    count:= count+1;
    A[count]:= n;
  fi
od:
seq(A[i], i=1..count); # Robert Israel, Nov 04 2015

Mathematica

Flatten[Position[Rest[FoldList[Times,1,Prime[Range[180]]]]+1,?PrimeQ]] (* _Harvey P. Dale, May 04 2012 *) (* this program generates the first 9 positive terms of the sequence; changing the Range constant to 33237 will generate all 23 terms above, but it will take a long time to do so *)

PARI

is(n)=ispseudoprime(prod(i=1,n,prime(i))+1) \\ Charles R Greathouse IV, Mar 21 2013

PARI

P=1; n=0; forprime(p=1, 10^5, if(ispseudoprime(P+1), print1(n", ")); n=n+1; P*=p;) \\ Hans Loeblich, May 10 2019

Formula

a(n+1) = A000720(A005234(n)). - M. F. Hasler, May 31 2018

Extensions

More terms from Labos Elemer

a(21) from Arlin Anderson (starship1(AT)gmail.com), Oct 20 2000

a(22)-a(23) from Eric W. Weisstein, Mar 13 2004 (based on information in A057704)

Offset and first term changed by Altug Alkan, Nov 27 2015

a(24) from Jeppe Stig Nielsen, Aug 08 2024

a(25) from Jeppe Stig Nielsen, Sep 01 2024

a(26) from Jeppe Stig Nielsen, Sep 24 2024

a(27) from Jeppe Stig Nielsen, Nov 10 2024

a(28) from Jeppe Stig Nielsen, Aug 21 2025