%I #15 Jul 11 2018 06:42:11

%S 907,5611,4318,26914,12238,76414,34738,138913,555613,2222413,13890013,

%T 55560013,222240013,1389000013,5556000013,22224000013,138900000013,

%U 555600000013,2222400000013,13890000000013,55560000000013,222240000000013,1389000000000013,5556000000000013,22224000000000013

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

%C 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 907/16 gives 56 with remainder 11;

%e 5611/13 gives 431 with remainder 8;

%e 4318/16 gives 269 with remainder 14;

%e 26914/22 gives 122 with remainder 38;

%e . . .

%e Now from 2222413 on, starts a devilish 0-inflation "from the middle" in a ternary cycle:

%e 2222413

%e 13890013

%e 55560013

%e 222240013

%e 1389000013

%e 5556000013

%e 22224000013

%e 138900000013

%e 555600000013

%e 2222400000013

%e 13890000000013

%e 55560000000013

%e 222240000000013

%e 1389000000000013

%e 5556000000000013

%e 22224000000000013

%e 138900000000000013

%e 555600000000000013

%e 2222400000000000013

%e . . .

%e We have:

%e 1389(k zeros)13

%e 5556(k zeros)13

%e 22224(k zeros)13

%e then:

%e 1389(k+2 zeros)13

%e 5556(k+2 zeros)13

%e 22224(k+2 zeros)13

%e then:

%e 1389(k+4 zeros)13

%e 5556(k+4 zeros)13

%e 22224(k+4 zeros)13

%e Etc.

%t NestList[FromDigits@ Flatten[IntegerDigits@ # & /@ QuotientRemainder[#, Total[IntegerDigits@ #]]] &, 907, 24] (* _Michael De Vlieger_, Jul 10 2018 *)

%Y Cf. A316650 (where the rule is explained) and A316680 (for the number 1358 that generates a similar pattern).

%K base,nonn

%O 1,1

%A _Eric Angelini_ and _Jean-Marc Falcoz_, Jul 10 2018