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A316679 The integer 907 and its infinite growing pattern (when iterating the rule explained in A316650 and hereunder, in the Comment section).

%I A316679 #15 Jul 11 2018 06:42:11
%S A316679 907,5611,4318,26914,12238,76414,34738,138913,555613,2222413,13890013,
%T A316679 55560013,222240013,1389000013,5556000013,22224000013,138900000013,
%U A316679 555600000013,2222400000013,13890000000013,55560000000013,222240000000013,1389000000000013,5556000000000013,22224000000000013
%N A316679 The integer 907 and its infinite growing pattern (when iterating the rule explained in A316650 and hereunder, in the Comment section).
%C A316679 It is conjectured, when iterating the idea explained in A316650 ("Result when n is divided by the sum of its digits and the resulting integer is concatenated to the remainder"), that all integers will end either on a fixed point (the first ones are listed in A052224) or grow forever (like 907 or 1358).
%e A316679 907/16 gives 56 with remainder 11;
%e A316679 5611/13 gives 431 with remainder 8;
%e A316679 4318/16 gives 269 with remainder 14;
%e A316679 26914/22 gives 122 with remainder 38;
%e A316679 . . .
%e A316679 Now from 2222413 on, starts a devilish 0-inflation "from the middle" in a ternary cycle:
%e A316679 2222413
%e A316679 13890013
%e A316679 55560013
%e A316679 222240013
%e A316679 1389000013
%e A316679 5556000013
%e A316679 22224000013
%e A316679 138900000013
%e A316679 555600000013
%e A316679 2222400000013
%e A316679 13890000000013
%e A316679 55560000000013
%e A316679 222240000000013
%e A316679 1389000000000013
%e A316679 5556000000000013
%e A316679 22224000000000013
%e A316679 138900000000000013
%e A316679 555600000000000013
%e A316679 2222400000000000013
%e A316679 . . .
%e A316679 We have:
%e A316679 1389(k zeros)13
%e A316679 5556(k zeros)13
%e A316679 22224(k zeros)13
%e A316679 then:
%e A316679 1389(k+2 zeros)13
%e A316679 5556(k+2 zeros)13
%e A316679 22224(k+2 zeros)13
%e A316679 then:
%e A316679 1389(k+4 zeros)13
%e A316679 5556(k+4 zeros)13
%e A316679 22224(k+4 zeros)13
%e A316679 Etc.
%t A316679 NestList[FromDigits@ Flatten[IntegerDigits@ # & /@ QuotientRemainder[#, Total[IntegerDigits@ #]]] &, 907, 24] (* _Michael De Vlieger_, Jul 10 2018 *)
%Y A316679 Cf. A316650 (where the rule is explained) and A316680 (for the number 1358 that generates a similar pattern).
%K A316679 base,nonn
%O A316679 1,1
%A A316679 _Eric Angelini_ and _Jean-Marc Falcoz_, Jul 10 2018