A088411 -id:A088411 - cp's OEIS frontend

1, 1, 2, 2, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Frank Ellermann, Mar 06 2020

nonn

a(n) = 0 marks a prime gap size of at least 2*prime(n+1)-1, e.g., primorial(8) +- prime(9) = {9699667,9699713} are primes, gap 2*23-1.
Mathworld reports that it is not known if there are an infinite number of prime Euclid numbers.
The tables in Ondrejka's collection contain no further primorial twin primes after {2309,2311} = primorial(13) +- 1 up to primorial(15877) +- 1 with 6845 digits.

			a(2) = a(3) = a(5) = 2: 2*3 +-1 = {5,7}, 6*5 +-1 = {29,31} and 210*11 +-1 = {2309,2311} are twin primes.
a(1) = a(4) = a(6) = 1: 1, 30*7 - 1 = 209 and 2310*13 + 1 = 30031 are not primes.
a(7) = 0: 510509 = 61 * 8369 and 510511 = 19 * 26869 are not primes.

H. Dubner, A new primorial prime, J. Rec. Math., 21 (No. 4, 1989), 276.

Chris K. Caldwell, the top 20: Primorial, 2012.
H. Dubner & N. J. A. Sloane, Correspondence, 1991, on A005234.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 30029.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 9699667.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations, 2001, tables 20, 20A, 20B.
Eric Weisstein's World of Mathematics, Primorial Prime.
Eric Weisstein's World of Mathematics, Euclid Number.

Cf. A096831, A002110 (primorials, p#), A057706.
Cf. A006862 (Euclid, p#+1), A005234 (prime p#+1), A014545 (index prime p#+1).
Cf. A057588 (Kummer, p#-1), A006794 (prime p#-1), A057704 (index prime p#-1).
Cf. A010051, A088411 (where a(n) is positive), A088257.

p:= proc(n) option remember; `if`(n<1, 1, ithprime(n)*p(n-1)) end:
a:= n-> add(`if`(isprime(p(n)+i), 1, 0), i=[-1, 1]):
seq(a(n), n=0..120);  # Alois P. Heinz, Mar 18 2020

primorial[n_] := primorial[n] = Times @@ Prime[Range[n]];
a[n_] := Boole@PrimeQ[primorial[n] - 1] + Boole@PrimeQ[primorial[n] + 1];
a /@ Range[0, 105] (* Jean-François Alcover, Nov 30 2020 *)

S = ''                     ;  Q = 1
do N = 1 to 27
   Q = Q * PRIME( N )
   T = ISPRIME( Q - 1 ) + ISPRIME( Q + 1 )
   S = S || ',' T
end N
S = substr( S, 3 )
say S                      ;  return S

a(n) = [ isprime(primorial(n) - 1) ] + [ isprime(primorial(n) + 1) ].

a(n) = Sum_{i in {-1,1}} A010051(primorial(n) + i).

cp's OEIS Frontend

A333058 0, 1, or 2 primes at primorial(n) +- 1.

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