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A279076 Maximum starting value of X such that repeated replacement of X with X-ceiling(X/6) requires n steps to reach 0.

%I A279076 #13 Mar 20 2022 18:26:24
%S A279076 0,1,2,3,4,5,7,9,11,14,17,21,26,32,39,47,57,69,83,100,121,146,176,212,
%T A279076 255,307,369,443,532,639,767,921,1106,1328,1594,1913,2296,2756,3308,
%U A279076 3970,4765,5719,6863,8236,9884,11861,14234,17081,20498,24598,29518,35422
%N A279076 Maximum starting value of X such that repeated replacement of X with X-ceiling(X/6) requires n steps to reach 0.
%C A279076 Inspired by A278586.
%C A279076 Limit_{n->oo} a(n)/(6/5)^n = 3.24387249751177521384734853905517802618171089570674...
%H A279076 A.H.M. Smeets, <a href="/A279075/a279075.txt">Constants related to repeated replacement of X with X-ceiling(X/d)</a>
%F A279076 a(n) = floor(a(n-1)*6/5) + 1.
%e A279076 7 -> 7-ceiling(7/6) = 5,
%e A279076 5 -> 5-ceiling(5/6) = 4,
%e A279076 4 -> 4-ceiling(4/6) = 3,
%e A279076 3 -> 3-ceiling(3/6) = 2,
%e A279076 2 -> 2-ceiling(2/6) = 1,
%e A279076 1 -> 1-ceiling(1/6) = 0,
%e A279076 so reaching 0 from 7 requires 6 steps;
%e A279076 8 -> 8-ceiling(8/6) = 6,
%e A279076 6 -> 6-ceiling(6/6) = 5,
%e A279076 5 -> 5-ceiling(5/6) = 4,
%e A279076 4 -> 4-ceiling(4/6) = 3,
%e A279076 3 -> 3-ceiling(3/6) = 2,
%e A279076 2 -> 2-ceiling(2/6) = 1,
%e A279076 1 -> 1-ceiling(1/6) = 0,
%e A279076 so reaching 0 from 8 (or more) requires 7 (or more) steps;
%e A279076 thus, 7 is the largest starting value from which 0 can be reached in 6 steps, so a(6) = 7.
%o A279076 (Magma) a:=[0]; aCurr:=0; for n in [1..51] do aCurr:=Floor(aCurr*6/5)+1; a[#a+1]:=aCurr; end for; a;
%Y A279076 Cf. A278586.
%Y A279076 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), (this sequence) (k=6), A279077 (k=7), A279078 (k=8), A279079 (k=9), A279080 (k=10). For each of these values of k, is the sequence the L-sieve transform of {k-1, 2k-1, 3k-1, ...}?
%K A279076 nonn
%O A279076 0,3
%A A279076 _Jon E. Schoenfield_, Dec 06 2016