%I #24 Sep 27 2017 12:17:02
%S 1,2,3,4,5,6,7,8,9,10,12,14,16,18,20,29,38,40,49,58,60,69,78,80,89,98,
%T 100,199,399,599,799,999,2999,4999,6999,8999,19999,39999,59999,79999,
%U 99999,299999,499999,699999,899999,1999999,3999999,5999999,7999999,9999999
%N A number N is called a "rebel" number if there do not exist two integers a and b such that N = a + b with a > b > 0 and S(a) = S(b) where S(n) is the sum of the digits of the number n.
%C A number which is not a "rebel" is called "docile". These definitions come from the French site Diophante, see link.
%C There are an infinite number of odd "rebel" numbers. For instance, all the repdigits 9999999...99 with a string of (2k+1) times the digit 9; but there are only eighteen even "rebel" numbers: {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 38, 40, 58, 60, 78, 80, 98, 100}.
%H Maurice Bauval, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a3-nombres-remarquables/2948-a348-dociles-et-rebelles">Dociles et Rebelles</a>, Diophante, A 348, September 2014.
%e 14 is rebel because 14 = 13 + 1 = 12 + 2 = 11 + 3 = 10 + 4 = 9 + 5 = 8 + 6 and never S(a) = S(b) with these integers.
%t Select[Range@ 100, Count[IntegerPartitions[#, {2}], _?(And[#1 > #2, Total@ IntegerDigits@ #1 == Total@ IntegerDigits@ #2] & @@ # &)] == 0 &]~Join~Union@ Flatten@ Table[Map[FromDigits[{#}~Join~ConstantArray[9, k]] &, Range[1, 9, 2] - Boole[OddQ@ k]], {k, 2, 6}] (* _Michael De Vlieger_, Sep 18 2017 *)
%o (PARI) isok(n) = {for (x=1, n\2, if ((x != (n-x)) && (sumdigits(x) == sumdigits(n-x)), return (0));); return (1);} \\ _Michel Marcus_, Sep 18 2017
%Y Cf. A292513 ("docile" numbers).
%K nonn,base
%O 1,2
%A _Bernard Schott_, Sep 18 2017
%E More terms from _Giovanni Resta_, Sep 18 2017