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

%I A279097 #15 Mar 29 2017 19:43:23
%S A279097 1,2,4,8,11,17,18,21,25,32,34,35,39,40,42,47,48,58,59,63,65,66,67,69,
%T A279097 90,91,97,105,110,122,140,144,151,152,162,166,168,173,174,175,177,179,
%U A279097 180,186,205,207,208,210,211,218,221,233,243,249,256,260,261,262
%N A279097 Numbers k such that prime(k) divides primorial(j) + 1 for some j.
%C A279097 As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).
%C A279097 Primorial(j) + 1 is the j-th Euclid number, A006862(j).
%H A279097 Giovanni Resta, <a href="/A279097/b279097.txt">Table of n, a(n) for n = 1..10000</a>
%e A279097 1 is in the sequence because primorial(0) + 1 = 1 + 1 = 2 is divisible by prime(1) = 2.
%e A279097 4 is in the sequence because primorial(2) + 1 = 2*3 + 1 = 7 is divisible by prime(4) = 7.
%e A279097 8 is in the sequence because primorial(7) + 1 = 2*3*5*7*11*13*17 + 1 = 510511 is divisible by prime(8) = 19.
%e A279097 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).
%e A279097 5 is not in the sequence because there is no number j such that primorial(j) + 1 is divisible by prime(5) = 11:
%e A279097     primorial(1) + 1 = 2       + 1 =   3 == 3 (mod 11)
%e A279097     primorial(2) + 1 = 2*3     + 1 =   7 == 7 (mod 11)
%e A279097     primorial(3) + 1 = 2*3*5   + 1 =  31 == 9 (mod 11)
%e A279097     primorial(4) + 1 = 2*3*5*7 + 1 = 211 == 2 (mod 11)
%e A279097 and primorial(j) + 1 = 2*...*11*... + 1  == 1 (mod 11) for all j >= 5.
%t A279097 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 *)
%Y A279097 Cf. A000040, A002110, A006862, A113165, A279098, A279099, A283928.
%K A279097 nonn
%O A279097 1,2
%A A279097 _Jon E. Schoenfield_, Mar 24 2017