
COMMENTS

The sequence is finite, more specifically there cannot be more than 19 terms. Proof: To have p = k*R(p) /+ q, with q < p, we must have 0 < k = (p +/ q) / R(p) < 2p / (p/10) = 20, since the prime p cannot end in 0 and therefore R(p) > p/10. (R(p) and p have the same number of digits.) Thus, for a given prime p, there cannot be more than 19 solutions (k=1,...,19) to p  k*R(p) = +/ q, and therefore no a(n) beyond n=19.  M. F. Hasler, Mar 13 2012, improved following remarks from Hans Havermann, Mar 14 2012
A refined analysis shows that the maximal number is less. On one hand, if R(p) is odd, then only even k can yield a prime. Therefore the maximal number of solutions can only be obtained for p starting with an even digit <= 8, and therefore p/R(p) < 9, thus k < 2p/R(p) < 18, k <= 17. Moreover, R(p) is not a multiple of 3 (since p isn't), therefore 1/3 of the kvalues lead to q=0 (mod 3) and are excluded, which leaves at most 11 possibilities. Other kvalues lead to q=0 (mod 5), unless p starts with '5'. In any case there cannot be more than 9 solutions.  M. F. Hasler, Mar 14 2012
a(9) > 8*10^15, if it exists.  Giovanni Resta, Oct 29 2018


PROG

(PARI) A202286(n)={ forprime(p=1, default(primelimit), my(r=A004086(p)); 2*p > n*r & sum(k=1, (2*p1)\r, isprime(abs(pk*r)))==n & return(p))} \\  M. F. Hasler, Mar 14 2012
