A279097 Numbers k such that prime(k) divides primorial(j) + 1 for some j.

1, 2, 4, 8, 11, 17, 18, 21, 25, 32, 34, 35, 39, 40, 42, 47, 48, 58, 59, 63, 65, 66, 67, 69, 90, 91, 97, 105, 110, 122, 140, 144, 151, 152, 162, 166, 168, 173, 174, 175, 177, 179, 180, 186, 205, 207, 208, 210, 211, 218, 221, 233, 243, 249, 256, 260, 261, 262

Offset: 1

Author

Jon E. Schoenfield, Mar 24 2017

Keywords

• nonn

Comments

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).

Primorial(j) + 1 is the j-th Euclid number, A006862(j).

Examples

			1 is in the sequence because primorial(0) + 1 = 1 + 1 = 2 is divisible by prime(1) = 2.
4 is in the sequence because primorial(2) + 1 = 2*3 + 1 = 7 is divisible by prime(4) = 7.
8 is in the sequence because primorial(7) + 1 = 2*3*5*7*11*13*17 + 1 = 510511 is divisible by prime(8) = 19.
59 is in the sequence because primorial(7) + 1 = 510511 is divisible by prime(59) = 277 (and primorial(17) + 1 = 1922760350154212639071 is divisible by prime(59) as well).
5 is not in the sequence because there is no number j such that primorial(j) + 1 is divisible by prime(5) = 11:
    primorial(1) + 1 = 2       + 1 =   3 == 3 (mod 11)
    primorial(2) + 1 = 2*3     + 1 =   7 == 7 (mod 11)
    primorial(3) + 1 = 2*3*5   + 1 =  31 == 9 (mod 11)
    primorial(4) + 1 = 2*3*5*7 + 1 = 211 == 2 (mod 11)
and primorial(j) + 1 = 2*...*11*... + 1  == 1 (mod 11) for all j >= 5.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A002110, A006862, A113165, A279098, A279099, A283928.

Programs

Mathematica

np[1]=1; np[n_] := Block[{c=0, p=Prime[n], trg, x=1}, trg = p-1; Do[x = Mod[x Prime[k], p]; If[trg == x, c++], {k, n-1}]; c]; Select[Range[262], np[#] > 0 &] (* Giovanni Resta, Mar 29 2017 *)