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

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

0,3

Comments

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.

References

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

Links

Table of n, a(n) for n=0..105.

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.

Formula

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

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

(Rexx)
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

Crossrefs

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.

Sequence in context: A305615 A038540 A335060 * A084348 A210580 A284478

Adjacent sequences: A333055 A333056 A333057 * A333059 A333060 A333061

Keyword

nonn

Author

Frank Ellermann, Mar 06 2020

Status

approved