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A279077 Maximum starting value of X such that repeated replacement of X with X-ceiling(X/7) requires n steps to reach 0.

%I A279077 #9 Mar 20 2022 18:26:30
%S A279077 0,1,2,3,4,5,6,8,10,12,15,18,22,26,31,37,44,52,61,72,85,100,117,137,
%T A279077 160,187,219,256,299,349,408,477,557,650,759,886,1034,1207,1409,1644,
%U A279077 1919,2239,2613,3049,3558,4152,4845,5653,6596,7696,8979,10476,12223,14261
%N A279077 Maximum starting value of X such that repeated replacement of X with X-ceiling(X/7) requires n steps to reach 0.
%C A279077 Inspired by A278586.
%C A279077 Limit_{n->oo} a(n)/(7/6)^n = 4.03710211215303193642791458111196922950551168987041...
%F A279077 a(n) = floor(a(n-1)*7/6) + 1.
%e A279077   10 -> 10-ceiling(10/7) = 8,
%e A279077    8 ->  8-ceiling(8/7)  = 6,
%e A279077    6 ->  6-ceiling(6/7)  = 5,
%e A279077    5 ->  5-ceiling(5/7)  = 4,
%e A279077    4 ->  4-ceiling(4/7)  = 3,
%e A279077    3 ->  3-ceiling(3/7)  = 2,
%e A279077    2 ->  2-ceiling(2/7)  = 1,
%e A279077    1 ->  1-ceiling(1/7)  = 0,
%e A279077 so reaching 0 from 10 requires 8 steps;
%e A279077   11 -> 11-ceiling(11/7) = 9,
%e A279077    9 ->  9-ceiling(9/7)  = 7,
%e A279077    7 ->  7-ceiling(7/7)  = 6,
%e A279077    6 ->  6-ceiling(6/7)  = 5,
%e A279077    5 ->  5-ceiling(5/7)  = 4,
%e A279077    4 ->  4-ceiling(4/7)  = 3,
%e A279077    3 ->  3-ceiling(3/7)  = 2,
%e A279077    2 ->  2-ceiling(2/7)  = 1,
%e A279077    1 ->  1-ceiling(1/7)  = 0,
%e A279077 so reaching 0 from 11 (or more) requires 9 (or more) steps;
%e A279077 thus, 10 is the largest starting value from which 0 can be reached in 8 steps, so a(8) = 10.
%o A279077 (Magma) a:=[0]; aCurr:=0; for n in [1..53] do aCurr:=Floor(aCurr*7/6)+1; a[#a+1]:=aCurr; end for; a;
%Y A279077 Cf. A278586.
%Y A279077 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), (this sequence) (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 A279077 nonn
%O A279077 0,3
%A A279077 _Jon E. Schoenfield_, Dec 06 2016