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

907, 5611, 4318, 26914, 12238, 76414, 34738, 138913, 555613, 2222413, 13890013, 55560013, 222240013, 1389000013, 5556000013, 22224000013, 138900000013, 555600000013, 2222400000013, 13890000000013, 55560000000013, 222240000000013, 1389000000000013, 5556000000000013, 22224000000000013

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Jul 10 2018

Keywords

• base
• nonn

Comments

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).

Examples

			907/16 gives 56 with remainder 11;
5611/13 gives 431 with remainder 8;
4318/16 gives 269 with remainder 14;
26914/22 gives 122 with remainder 38;
. . .
Now from 2222413 on, starts a devilish 0-inflation "from the middle" in a ternary cycle:
2222413
13890013
55560013
222240013
1389000013
5556000013
22224000013
138900000013
555600000013
2222400000013
13890000000013
55560000000013
222240000000013
1389000000000013
5556000000000013
22224000000000013
138900000000000013
555600000000000013
2222400000000000013
. . .
We have:
1389(k zeros)13
5556(k zeros)13
22224(k zeros)13
then:
1389(k+2 zeros)13
5556(k+2 zeros)13
22224(k+2 zeros)13
then:
1389(k+4 zeros)13
5556(k+4 zeros)13
22224(k+4 zeros)13
Etc.
		

Crossrefs

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

Programs

Mathematica

NestList[FromDigits@ Flatten[IntegerDigits@ # & /@ QuotientRemainder[#, Total[IntegerDigits@ #]]] &, 907, 24] (* Michael De Vlieger, Jul 10 2018 *)