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A279080 Maximum starting value of X such that repeated replacement of X with X-ceiling(X/10) requires n steps to reach 0. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Inspired by A278586.

Lim_{n->inf} 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: A218470 A008727 A088450 * A108641 A289351 A171550

Adjacent sequences:  A279077 A279078 A279079 * A279081 A279082 A279083

KEYWORD

nonn

AUTHOR

Jon E. Schoenfield, Dec 06 2016

STATUS

approved

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Last modified September 27 18:40 EDT 2020. Contains 337386 sequences. (Running on oeis4.)