A118739 Let L_n be the infinite sequence formed by starting with 1 and repeatedly placing the first digit at the end of the number and adding n to get the next term. If L_n eventually reaches a cycle, sequence gives length of that cycle, otherwise -1.

9, 9, 3, 9, 18, 3, 18, 9, 1, 18, 9, 6, 9, 18, 6, 18, 9, 1, 18, 36, 9, 18, 18, 9, 18, 9, 3, 18, 36, 9, 18, 27, 3, 18, 9, 3, 45, 36, 6, 9, 27, 12, 27, 18, 3, 72, 36, 9, 9, 27, 9, 27, 18, 3, 99, 27, 9, 45, 27, 9, 27, 18, 3, 27, 27, 9, 45, 27, 9, 27, 18, 3, 27, 27, 9, 54, 27, 9, 27, 36, 3, 27

Offset

1,1

Comments

It is conjectured that L_n always reaches a cycle.

From Robert Israel, Dec 26 2017: (Start)

It always reaches a cycle. Suppose d = A055642(n).

If L_n(k) < 10^(d+1) + 10^d, then L_n(k+1) < 10^(d+1) + n < 10^(d+1) + 10^d. Thus the sequence L_n is bounded, and always reaches a cycle.

Moreover, a(n) <= 11*10^A055642(n). (End)

From Robert Israel, Dec 25 2017: (Start)

a(10^k-1) = 1 for k >= 1.

a(19*10^k-1) = 1.

Empirical:

a(10^k) = 9*(k+1).

a(2*10^k) = 9*(k+1)^2.

a(2*10^k-1) = 9*(k+1).

a(10^k+1) = 9*(k+1) for k >= 2.

a(2*10^k+1) = 3*(k+2) for k >= 1. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

L_5 = [1,6,11,16,66,71,22,27,77,82,33,38,88,93,44,49,99,104,46,69,101,16,...]
enters a cycle of length 18 after 3 steps.

Maple

h:= proc(n) local t, k, S, d;
    t:= 1; S[t]:= 0;
    for k from 1 do
      d:= 10^ilog10(t);
      t:= 10*(t mod d)+ floor(t/d) + n;
      if assigned(S[t]) then return k-S[t] fi;
      S[t]:= k;
    od
end proc:
map(h, [$1..100]); # Robert Israel, Dec 25 2017

Mathematica

h[n_] := Module[{t, k, S, d}, t = 1; S[_] = 0; For[k = 1, True, k++, d = 10^Floor[Log[10, t]]; t = 10*Mod[t, d] + Floor[t/d] + n; If[S[t] != 0, Return[k - S[t]]]; S[t] = k]];
Array[h, 100] (* Jean-François Alcover, Dec 26 2017, after Robert Israel *)

Crossrefs

Cf. A055642, A118740.

Sequence in context: A180012 A172504 A172502 * A192031 A283749 A249023

Adjacent sequences: A118736 A118737 A118738 * A118740 A118741 A118742

Keyword

base,nonn

Author

Luc Stevens (lms022(AT)yahoo.com), May 24 2006

Extensions

Corrected by Robert Israel, Dec 25 2017

Status

approved