A316680 -id:A316680 - cp's OEIS frontend

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

Eric Angelini and Jean-Marc Falcoz, Jul 10 2018

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).

			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.

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

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

A316678 Smallest numbers leading in n steps to a term that repeats itself, according to the rule explained in A316650 (and hereunder in the Comment section).

Original entry on oeis.org

19, 31, 13, 16, 32, 11, 23, 15, 236, 282, 341, 1047, 787, 419, 286, 626, 557, 498, 1357, 1001, 368, 1921, 917, 2077, 3319, 3457, 5090, 2294, 2144, 3501, 4485, 10661, 16753, 3092, 5252, 3475, 2102, 3572, 656, 1691, 7461, 10445, 4596, 13937, 15964, 25540, 14380

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jul 10 2018

The Crossrefs section gives two more interesting such infinite growing patterns.

			19 is the smallest integer leading to itself in 1 step because we have [19/10 = 10*1 + 9];
31 is the smallest integer ending on a fixed point in 2 steps because 31 leads to 73 [31/4 = 4*7 + 3] (step 1) and 73 to itself [73/10 = 10*7 + 3] (step 2);
13 is the smallest integer ending on a fixed point in 3 steps because 13 leads to 31 [13/4 = 4*3 + 1] (step 1) and 31 leads to 73 in 2 steps (see above);
16 is the smallest integer ending on a fixed point in 4 steps because 16 leads to 22 [16/7 = 7*2 + 2] (step 1), then 22 leads to 52 [22/4 = 4*5+2] (step 2), then 52 leads to 73 [52/7 = 7*7 + 3] (step 3) and 73 to itself [73/10 = 10*7 + 3] (step 4);
32 is the smallest integer ending on a fixed point in 5 steps [32,62,76,511,730];
11 is the smallest integer ending on a fixed point in 6 steps
  [11,51,83,76,511,730];
23 is the smallest integer ending on a fixed point in 7 steps
  [23,43,61,85,67,52,73];
Etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..301

Cf. A316650 (where the main idea is explained), A316679 (for the infinite growing pattern produced by 907) and A316680 (for the infinite growing pattern produced by 1358).

Array[Block[{k = 1}, While[Count[#, 0] != 1 &@ Differences@ NestList[FromDigits@ Flatten[IntegerDigits@ # & /@ QuotientRemainder[#, Total[IntegerDigits@ #]]] &, k, #], k++]; k] &, 47] (* Michael De Vlieger, Jul 10 2018 *)

cp's OEIS Frontend

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

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