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

Offset

1,1

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

Links

Table of n, a(n) for n=1..21.

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

Formula

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

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

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A098833 A075371 A218948 * A281382 A078203 A285937

Adjacent sequences: A057701 A057702 A057703 * A057705 A057706 A057707

Keyword

nonn,more

Author

Labos Elemer, Oct 24 2000

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

Status

approved