Cinquante signes

  (Dall-e creation ) To my surprise,  S  doesn't seem to be in the OEIS. Each term of  S  is the concatenation of n and its binary expansion . Example: we form  971100001  by concatenating the decimal number 97 to its binary expansion 1100001 . S  was simply computed using the first 666 lines of  this table  (not counting the 0 0 line). We could call such numbers “Enbi numbers” [ n  + bi ( nary ); a  few questions  about Enbi numbers after the hereunder  S . ] S = 11, 210, 311, 4100, 5101, 6110, 7111, 81000, 91001, 101010, 111011, 121100, 131101, 141110, 151111, 1610000, 1710001, 1810010, 1910011, 2010100, 2110101, 2210110, 2310111, 2411000, 2511001, 2611010, 2711011, 2811100, 2911101, 3011110, 3111111, 32100000, 33100001, 34100010, 35100011, 36100100, 37100101, 38100110, 39100111, 40101000, 41101001, 42101010, 43101011, 44101100, 45101101, 46101110, 47101111, 48110000, 49110001, 50110010, 51110011, 52110100, 53110...

Joy Division Disney Division Take 2024 for instance. Divide all even digits by 2 and iterate until it's no longer possible. 2024 --> 1012 --> 1011 We says that 2024 ends in a binary string (with decimal equivalent 11 here). If the successive divisions don't end in a binary string, we multiply the end result by 3 and iterate. This is the start of the journey of a(1) = 1 (divisions in yellow): S = 1, 3, 9, 2 7, 17, 51, 153, 4 59, 159, 4 77, 177, 531, 1593, 4 799, 1779, 5337, 1 6 011, 13011,39033, 117099, 351 2 97, 351197, etc. I've computed  S  by hand (with my iPhone)...  and stopped after reaching 111773931539... (more terms by Giorgos Kalogeropoulos in the "Easter  update #2" section, below) Question Do all integers end sooner or later in a binary string? I guess some journeys might be a bit too long to check! (work in progress — leaving home now for Boris ' birthday party!-) ____________________ Easter  update #1 Jean-Marc Falcoz >  Parmi le...