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

0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 26, 31, 37, 44, 52, 61, 72, 85, 100, 117, 137, 160, 187, 219, 256, 299, 349, 408, 477, 557, 650, 759, 886, 1034, 1207, 1409, 1644, 1919, 2239, 2613, 3049, 3558, 4152, 4845, 5653, 6596, 7696, 8979, 10476, 12223, 14261

Offset

0,3

Comments

Inspired by A278586.

Limit_{n->oo} a(n)/(7/6)^n = 4.03710211215303193642791458111196922950551168987041...

Links

Table of n, a(n) for n=0..53.

Formula

a(n) = floor(a(n-1)*7/6) + 1.

Example

  10 -> 10-ceiling(10/7) = 8,
   8 ->  8-ceiling(8/7)  = 6,
   6 ->  6-ceiling(6/7)  = 5,
   5 ->  5-ceiling(5/7)  = 4,
   4 ->  4-ceiling(4/7)  = 3,
   3 ->  3-ceiling(3/7)  = 2,
   2 ->  2-ceiling(2/7)  = 1,
   1 ->  1-ceiling(1/7)  = 0,
so reaching 0 from 10 requires 8 steps;
  11 -> 11-ceiling(11/7) = 9,
   9 ->  9-ceiling(9/7)  = 7,
   7 ->  7-ceiling(7/7)  = 6,
   6 ->  6-ceiling(6/7)  = 5,
   5 ->  5-ceiling(5/7)  = 4,
   4 ->  4-ceiling(4/7)  = 3,
   3 ->  3-ceiling(3/7)  = 2,
   2 ->  2-ceiling(2/7)  = 1,
   1 ->  1-ceiling(1/7)  = 0,
so reaching 0 from 11 (or more) requires 9 (or more) steps;
thus, 10 is the largest starting value from which 0 can be reached in 8 steps, so a(8) = 10.

Prog

(Magma) a:=[0]; aCurr:=0; for n in [1..53] do aCurr:=Floor(aCurr*7/6)+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), (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, ...}?

Sequence in context: A033058 A060470 A003044 * A018541 A361785 A361319

Adjacent sequences: A279074 A279075 A279076 * A279078 A279079 A279080

Keyword

nonn

Author

Jon E. Schoenfield, Dec 06 2016

Status

approved