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

Offset

1,3

References

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008.

Links

Table of n, a(n) for n=1..27.

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

Formula

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

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A280206 A190783 A136367 * A379046 A158930 A330263

Adjacent sequences: A014542 A014543 A014544 * A014546 A014547 A014548

Keyword

nonn,nice,hard,more

Author

Eric W. Weisstein, Murray R. Bremner

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

Status

approved