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%I #31 Feb 06 2024 21:48:28
%S 1,19,20,21,22,39,40,41,59,60,61,79,80,81,98,99,122,201,219,220
%N Numbers k such that it is possible to form a palindrome by concatenating the first k positive integers in some order.
%C For k=19 and k=20 there are 8! such palindromes.
%C There are no such palindromes for k = 101, 119, 121, 139, 141, 159, 161, 179, 181, 199. This is because it is impossible to have a pair matching "100".
%C It is unknown whether there exist palindromes that are a concatenation of the first k primes. The list of k for which such palindromes could exist is in S001070 (see Noe's link). It has been proven that k=36 and k=247 cannot exist.
%C For k=21, 11612160 admissible permutations produce 8069040 distinct palindromes. These counts are respectively equal to 20240640 and 13633200 for k=22. - _Giovanni Resta_, Sep 02 2017
%H Tony D. Noe, <a href="http://www.integersequences.org/s001070.html">S001070</a>
%H Tony D. Noe, <a href="http://www.integersequences.org/s001071.html">S001071</a>
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_891.htm">Puzzle 891: The first N integers arranged to form a Palindrome</a>, The Prime Puzzles and Problems Connection.
%e For k=19 one palindrome is 1.12.3.14.7.15.6.18.9.10.19.8.16.5.17.4.13.2.11 => 11231471561891019816517413211. This palindrome is the smallest such palindrome that is prime.
%Y Cf. A291633.
%K nonn,base,hard,more
%O 1,2
%A _Dmitry Kamenetsky_, Sep 02 2017