A057704 Primorial - 1 prime indices: integers m such that the m-th primorial minus 1 is prime.

2, 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, 234725, 334023, 435582, 446895

Offset: 1

Author

Labos Elemer, Oct 24 2000

Keywords

• nonn
• more

Comments

There are two versions of "primorial": this is using the definition in A002110. - Robert Israel, Dec 30 2014

As of 28 February 2012, the largest known primorial prime is A002110(85586) - 1 with 476311 digits, found by the PrimeGrid project (see link). - Dmitry Kamenetsky, Aug 11 2015

Examples

			The 6th primorial is A002110(6) = 2*3*5*7*11*13 = 30030, and 30030 - 1 = 30029 is a prime, so 6 is in the sequence.
		

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

Links

• Chris K. Caldwell, Prime Pages: Database Search
• Chris K. Caldwell, The top 20: primorial primes
• Sílvia Casacuberta, On the divisibility of binomial coefficients, arXiv:1906.07652 [math.NT], 2019. Mentions this sequence.
• Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola (2018) Vol. 54, Issue 3.
• Eric Weisstein's World of Mathematics, Integer Sequence Primes
• Eric Weisstein's World of Mathematics, Primorial Prime
• Wikipedia, Primorial prime

Crossrefs

Cf. A006794 (Primorial -1 primes: Primes p such that -1 + product of primes up to p is prime).

Cf. A057705, A014545, A005234, A002110, A057706.

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, Dec 25 2014

Mathematica

a057704[n_] :=
Flatten@Position[
Rest[FoldList[Times, 1, Prime[Range[n]]]] - 1, Integer?PrimeQ]; a057704[500] (* _Michael De Vlieger, Dec 25 2014 *)

PARI

lista(nn) = {s = 1; for(k=1, nn, s *= prime(k); if(ispseudoprime(s - 1), print1(k, ", ")); ); } \\ Altug Alkan, Dec 08 2015

PARI

is(n) = ispseudoprime(prod(k=1, n, prime(k)) - 1); \\ Altug Alkan, Dec 08 2015

Formula

a(n) = A000720(A006794(n)).

a(n) = primepi(A006794(n)).

Extensions

Corrected by Holzer Werner, Nov 28 2002

a(19)-a(20) from Eric W. Weisstein, Dec 08 2015 (Mark Rodenkirch confirms based on saved log files that all p < 700,000 have been tested)

a(21) from Jeppe Stig Nielsen, Oct 19 2021

a(22)-a(24) from Jeppe Stig Nielsen, Dec 16 2024