A279080 Maximum starting value of X such that repeated replacement of X with X-ceiling(X/10) requires n steps to reach 0.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 22, 25, 28, 32, 36, 41, 46, 52, 58, 65, 73, 82, 92, 103, 115, 128, 143, 159, 177, 197, 219, 244, 272, 303, 337, 375, 417, 464, 516, 574, 638, 709, 788, 876, 974, 1083, 1204, 1338, 1487, 1653, 1837, 2042, 2269

Offset

0,3

Comments

Inspired by A278586.

Limit_{n->oo} a(n)/(10/9)^n = 5.60655601136196116133057876294687807265035051745268...

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Formula

a(n) = floor(a(n-1)*10/9) + 1.

Example

  13 -> 13-ceiling(13/10) = 11,
  11 -> 11-ceiling(11/10) = 9,
   9 ->  9-ceiling(9/10)  = 8,
   8 ->  8-ceiling(8/10)  = 7,
...
   1 ->  1-ceiling(1/10)  = 0,
so reaching 0 from 13 requires 11 steps;
  14 -> 14-ceiling(14/10) = 12,
  12 -> 12-ceiling(12/10) = 10,
  10 -> 10-ceiling(10/10) = 9,
   9 ->  9-ceiling(9/10)  = 8,
   8 ->  8-ceiling(8/10)  = 7,
...
   1 ->  1-ceiling(1/10)  = 0,
so reaching 0 from 14 (or more) requires 12 (or more) steps;
thus, 13 is the largest starting value from which 0 can be reached in 11 steps, so a(11) = 13.

Maple

H:= proc(y) local u, v;
     v:= -y-1 mod 9+1;
     (10*y+v)/9
end proc:
A:= Array(0..100):
A[0]:= 0:
for i from 1 to 100 do A[i]:= H(A[i-1]) od:
convert(A, list); # Robert Israel, Jun 23 2020

Mathematica

With[{s = Array[-1 + Length@ NestWhileList[# - Ceiling[#/10] &, #, # > 0 &] &, 2400, 0]}, Array[-1 + Position[s, #][[-1, 1]] &, Max@ s, 0]] (* Michael De Vlieger, Jun 23 2020 *)

Prog

(Magma) a:=[0]; aCurr:=0; for n in [1..57] do aCurr:=Floor(aCurr*10/9)+1; a[#a+1]:=aCurr; end for; a;

Crossrefs

Cf. A278586.

See the following sequences for maximum starting value of X such that repeated replacement of X with X-ceiling(X/k) requires n steps to reach 0: A000225 (k=2), A006999 (k=3), A155167 (k=4, apparently; see Formula entry there), A279075 (k=5), A279076 (k=6), A279077 (k=7), A279078 (k=8), A279079 (k=9), (this sequence) (k=10). For each of these values of k, is the sequence the L-sieve transform of {k-1, 2k-1, 3k-1, ...}?

Sequence in context: A008727 A357745 A088450 * A108641 A366944 A289351

Adjacent sequences: A279077 A279078 A279079 * A279081 A279082 A279083

Keyword

nonn

Author

Jon E. Schoenfield, Dec 06 2016

Status

approved