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A279078 Maximum starting value of X such that repeated replacement of X with X-ceiling(X/8) requires n steps to reach 0.

%I A279078 #11 Mar 20 2022 18:26:41
%S A279078 0,1,2,3,4,5,6,7,9,11,13,15,18,21,25,29,34,39,45,52,60,69,79,91,105,
%T A279078 121,139,159,182,209,239,274,314,359,411,470,538,615,703,804,919,1051,
%U A279078 1202,1374,1571,1796,2053,2347,2683,3067,3506,4007,4580,5235,5983,6838
%N A279078 Maximum starting value of X such that repeated replacement of X with X-ceiling(X/8) requires n steps to reach 0.
%C A279078 Inspired by A278586.
%C A279078 Limit_{n->oo} a(n)/(8/7)^n = 4.42210347959393228709604412445802201220907917744900...
%F A279078 a(n) = floor(a(n-1)*8/7) + 1.
%e A279078   11 -> 11-ceiling(11/8) = 9,
%e A279078    9 ->  9-ceiling(9/8)  = 7,
%e A279078    7 ->  7-ceiling(7/8)  = 6,
%e A279078    6 ->  6-ceiling(6/8)  = 5,
%e A279078 ...
%e A279078    1 ->  1-ceiling(1/8)  = 0,
%e A279078 so reaching 0 from 11 requires 9 steps;
%e A279078   12 -> 12-ceiling(12/8) = 10,
%e A279078   10 -> 10-ceiling(10/8) = 8,
%e A279078    8 ->  8-ceiling(8/8)  = 7,
%e A279078    7 ->  7-ceiling(7/8)  = 6,
%e A279078 ...
%e A279078    1 ->  1-ceiling(1/8)  = 0,
%e A279078 so reaching 0 from 12 (or more) requires 10 (or more) steps;
%e A279078 thus, 11 is the largest starting value from which 0 can be reached in 9 steps, so a(9) = 11.
%o A279078 (Magma) a:=[0]; aCurr:=0; for n in [1..55] do aCurr:=Floor(aCurr*8/7)+1; a[#a+1]:=aCurr; end for; a;
%Y A279078 Cf. A278586.
%Y A279078 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), (this sequence) (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 A279078 nonn
%O A279078 0,3
%A A279078 _Jon E. Schoenfield_, Dec 06 2016