OFFSET
1,2
COMMENTS
The "Erase or triple" protocol describes how to transform an integer K into an integer L: if K has 2 or more identical digits, erase them to get L (1201331 becomes 20); if K has no duplicate digits, triple K to get L (20 becomes 60).
Some integers disappear immediately (like 11, 2002 or 1919188 - see the Crossref section), other enter into a loop if you apply this protocol to the successive results.
Note that 1102 is transformed into 2 because no leading zeroes are admitted.
It seems that all integers up to 10^6 enter into one of those four loops:
Loop zero: 0, 0, 0, 0, 0,...
Loop five: 5, 15, 45, 135, 405, 1215, 25, 75, 225, 5, ...
Loop seven: 7, 21, 63, 189, 567, 1701, 70, 210, 630, 1890, 5670, 17010, 7, ...
Loop eighty-nine : 89, 267, 801, 2403, 7209, 21627, 167, 501,1503, 4509, 13527, 40581, 121743, 2743, 8229, 89, ...
Since L <= 9876543210 if K > 9876543210, it suffices to only look at trajectories for 0 <= K <= 9876543210. Exhaustive search shows every trajectory converges to one of the four loops above. - Chai Wah Wu, Feb 11 2019
EXAMPLE
1 becomes 3 (the triple of 1), then 3 becomes 9 (the triple of 3), then 6 becomes 27 (the triple of 9) and now 81, 243, 729, 2187, 6561, 51 (because the pair 6-6 is erased), 153 (the triple of 54), ... until 17010 which will turn into a term already in the sequence (7). At this point the trajectory cycles.
MATHEMATICA
NestWhileList[Function[n, If[#1 == #2, 3 FromDigits@ #1, FromDigits@ #2] & @@ {#, Select[#, DigitCount[n, 10, #] == 1 &]} &@ IntegerDigits@ n], 1, Unequal, All] (* Michael De Vlieger, Feb 28 2018 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Eric Angelini and Jean-Marc Falcoz, Feb 27 2018
STATUS
approved