%N a(n) = smallest r where (concatenation of n, r times with itself)*10 + 1 is a prime given by A087403(n), or 0 if no such number exists.
%C Conjecture: No term is zero. [Warning: This is known to be wrong, see below. - _M. F. Hasler_, Jan 08 2015]
%C a(47), a(67), a(100), a(107), a(114) are zero or larger than 1000. - _Ray Chandler_, Sep 23 2003; edited by _M. F. Hasler_, Jan 08 2015
%C a(47) > 10000 or 0. a(67) > 10000 or 0. a(100) > 10000 or 0. a(107) = 2478. a(114) = 1164. See link for more details. - _Derek Orr_, Oct 02 2014
%C From _Farideh Firoozbakht_, Jan 07 2015: (Start)
%C The conjecture is not true and there exist many numbers n such that a(n)=0.
%C Theorem: If m is a positive integer and a(10^m)=r then r+1 divides m+1.
%C Corollary: If p is a prime number then a(10^(p-1))=0 or (10^(p^2)-1)/(10^p-1) is a prime number.
%C By using the theorem and its corollary we can prove that for m = 2, 3, ..., 275 a(10^m)=0.
%C What is the smallest odd prime p, such that (10^(p^2)-1)/(10^p-1) is a prime number (and a(10^(p-1)) could be nonzero)?
%C What is the smallest integer m > 1 such that a(10^m) is nonzero?
%C Conjecture: If n is not of the form 10^m then a(n) is nonzero.
%C _M. F. Hasler_ has checked proofs of the theorem and its corollary.
%H Derek Orr, <a href="/A086766/a086766.txt">Values of a(n) > 1000 for n < 1000</a>
%e a(2) = 3, 2221 is a prime but 21 and 221 are composite.
%o vector(46,n,a(n)) \\ _Derek Orr_, Oct 02 2014
%Y Cf. A087403.
%A _Amarnath Murthy_, Sep 10 2003
%E More terms from _Ray Chandler_, Sep 23 2003