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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.

%I A279080 #20 Mar 20 2022 18:26:18
%S A279080 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,
%T A279080 82,92,103,115,128,143,159,177,197,219,244,272,303,337,375,417,464,
%U A279080 516,574,638,709,788,876,974,1083,1204,1338,1487,1653,1837,2042,2269
%N A279080 Maximum starting value of X such that repeated replacement of X with X-ceiling(X/10) requires n steps to reach 0.
%C A279080 Inspired by A278586.
%C A279080 Limit_{n->oo} a(n)/(10/9)^n = 5.60655601136196116133057876294687807265035051745268...
%H A279080 Robert Israel, <a href="/A279080/b279080.txt">Table of n, a(n) for n = 0..10000</a>
%F A279080 a(n) = floor(a(n-1)*10/9) + 1.
%e A279080   13 -> 13-ceiling(13/10) = 11,
%e A279080   11 -> 11-ceiling(11/10) = 9,
%e A279080    9 ->  9-ceiling(9/10)  = 8,
%e A279080    8 ->  8-ceiling(8/10)  = 7,
%e A279080 ...
%e A279080    1 ->  1-ceiling(1/10)  = 0,
%e A279080 so reaching 0 from 13 requires 11 steps;
%e A279080   14 -> 14-ceiling(14/10) = 12,
%e A279080   12 -> 12-ceiling(12/10) = 10,
%e A279080   10 -> 10-ceiling(10/10) = 9,
%e A279080    9 ->  9-ceiling(9/10)  = 8,
%e A279080    8 ->  8-ceiling(8/10)  = 7,
%e A279080 ...
%e A279080    1 ->  1-ceiling(1/10)  = 0,
%e A279080 so reaching 0 from 14 (or more) requires 12 (or more) steps;
%e A279080 thus, 13 is the largest starting value from which 0 can be reached in 11 steps, so a(11) = 13.
%p A279080 H:= proc(y) local u,v;
%p A279080      v:= -y-1 mod 9+1;
%p A279080      (10*y+v)/9
%p A279080 end proc:
%p A279080 A:= Array(0..100):
%p A279080 A[0]:= 0:
%p A279080 for i from 1 to 100 do A[i]:= H(A[i-1]) od:
%p A279080 convert(A,list); # _Robert Israel_, Jun 23 2020
%t A279080 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 *)
%o A279080 (Magma) a:=[0]; aCurr:=0; for n in [1..57] do aCurr:=Floor(aCurr*10/9)+1; a[#a+1]:=aCurr; end for; a;
%Y A279080 Cf. A278586.
%Y A279080 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, ...}?
%K A279080 nonn
%O A279080 0,3
%A A279080 _Jon E. Schoenfield_, Dec 06 2016