

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).


1



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
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OFFSET

1,1


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 [10 is an example of a simple pattern: 10,100,1000,10000,100000,. . .]
The Crossrefs section gives two more interesting such infinite growing patterns.


LINKS



EXAMPLE

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.


MATHEMATICA

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 *)


CROSSREFS

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).


KEYWORD

base,nonn


AUTHOR



STATUS

approved



