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 A202286 Smallest prime which is equal to a multiple of its reversal +- a prime smaller than itself in exactly n ways. 4
 31, 41, 71, 61, 6421, 8501, 8116001, 845534401 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 k-values lead to q == 0 (mod 3) and are excluded, which leaves at most 11 possibilities. Other k-values 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 LINKS EXAMPLE a(4)=61 because 61 = 16 x 2 + 29, 61 = 16 x 3 + 13, 61 = 16 x 4 - 3, and 61 = 16 x 5 - 19. PROG (PARI) A202286(n)={ forprime(p=1, default(primelimit), my(r=A004086(p)); 2*p > n*r & sum(k=1, (2*p-1)\r, isprime(abs(p-k*r)))==n & return(p))}  \\ M. F. Hasler, Mar 14 2012 CROSSREFS Cf. A099180, A209063, A182239. Sequence in context: A089442 A243704 A060327 * A285805 A141180 A176371 Adjacent sequences:  A202283 A202284 A202285 * A202287 A202288 A202289 KEYWORD nonn,base,fini,more AUTHOR Claudio Meller, Dec 15 2011, a(3) from Olivier Gérard EXTENSIONS a(7)-a(8) from Hans Havermann, Mar 12 2012 STATUS approved

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Last modified March 31 13:07 EDT 2020. Contains 333151 sequences. (Running on oeis4.)