%I #29 Dec 15 2017 17:36:11
%S 2,4,6,8,22,33,44,55,66,77,88,99,111,141,171,222,232,242,252,262,272,
%T 282,292,414,424,434,444,454,464,474,484,494,606,616,626,636,646,656,
%U 666,676,686,696,717,747,777,828,838,848,858,868,878,888,898,2002,3113
%N Palindromes such that the GCD of any pair of successive terms is a palindromic prime.
%C From _Danny Rorabaugh_, Mar 21-24 2015: (Start)
%C This sequence is generated by setting a(1)=2 and letting a(n+1) be the least palindromic integer strictly greater than a(n) such that GCD(a(n),a(n+1)) is a palindromic prime.
%C By initiating with a different palindrome, a different sequence is generated: for example, [3, 6, 8, ...] or [151, 1661, 1771, 1881, 1991, 2002, 3113, ...]. 323 is the smallest palindrome greater than 1 whose sequence does not eventually coincide with the present sequence. Indeed, 323 is the smallest palindrome greater than 1 with no palindromic prime factor (thus, as with 1, the sequence beginning with 323 has only one term).
%C I conjecture that every sequence generated in this fashion, beginning with a palindrome that has a palindromic prime factor, is infinite and eventually coincides with this sequence.
%C (End)
%C The sequence, started with any palindrome having a palindromic prime factor, is certainly infinite: if a(n) is a palindrome divisible by palindromic prime p, then M = p*(1+10^m) for large enough m is another palindrome; if m is divisible by the order of 10 mod q for every prime q other than 2 and 5 dividing a(n), then gcd(M,a(n)) = p. - _Robert Israel_, Mar 24 2015
%H Danny Rorabaugh, <a href="/A082615/b082615.txt">Table of n, a(n) for n = 1..10000</a>
%o (PARI) ispal(n) = d = digits(n); Vecrev(d) == d;
%o lista(nn) = {prev = 2; print1(prev, ", "); for (n=2, nn, k = prev+1; while(!(ispal(k) && (p=gcd(prev, k)) && isprime(p) && ispal(p)), k++); print1(k, ", "); prev = k;);} \\ _Michel Marcus_, Mar 24 2015
%Y Cf. A002113, A082616.
%K easy,nonn,base
%O 1,1
%A _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 29 2003
%E Corrected and extended by _Jason Earls_, May 18 2004