cp's OEIS Frontend

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).

%I A316678 #17 Jul 11 2018 06:41:28
%S A316678 19,31,13,16,32,11,23,15,236,282,341,1047,787,419,286,626,557,498,
%T A316678 1357,1001,368,1921,917,2077,3319,3457,5090,2294,2144,3501,4485,10661,
%U A316678 16753,3092,5252,3475,2102,3572,656,1691,7461,10445,4596,13937,15964,25540,14380
%N 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).
%C A316678 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,. . .]
%C A316678 The Crossrefs section gives two more interesting such infinite growing patterns.
%H A316678 Jean-Marc Falcoz, <a href="/A316678/b316678.txt">Table of n, a(n) for n = 1..301</a>
%e A316678 19 is the smallest integer leading to itself in 1 step because we have [19/10 = 10*1 + 9];
%e A316678 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);
%e A316678 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);
%e A316678 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);
%e A316678 32 is the smallest integer ending on a fixed point in 5 steps [32,62,76,511,730];
%e A316678 11 is the smallest integer ending on a fixed point in 6 steps
%e A316678   [11,51,83,76,511,730];
%e A316678 23 is the smallest integer ending on a fixed point in 7 steps
%e A316678   [23,43,61,85,67,52,73];
%e A316678 Etc.
%t A316678 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 *)
%Y A316678 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).
%K A316678 base,nonn
%O A316678 1,1
%A A316678 _Eric Angelini_ and _Jean-Marc Falcoz_, Jul 10 2018