A279079 Maximum starting value of X such that repeated replacement of X with X-ceiling(X/9) requires n steps to reach 0.

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 19, 22, 25, 29, 33, 38, 43, 49, 56, 64, 73, 83, 94, 106, 120, 136, 154, 174, 196, 221, 249, 281, 317, 357, 402, 453, 510, 574, 646, 727, 818, 921, 1037, 1167, 1313, 1478, 1663, 1871, 2105, 2369, 2666, 3000, 3376, 3799

Offset

0,3

Comments

Inspired by A278586.

Lim_{n->inf} a(n)/(9/8)^n = 5.19544896392362185906460915572195169945039729234281...

Links

Table of n, a(n) for n=0..56.

Formula

a(n) = floor(a(n-1)*9/8) + 1.

Example

  12 -> 12-ceiling(12/9) = 10,
  10 -> 10-ceiling(10/9) = 8,
   8 ->  8-ceiling(8/9)  = 7,
   7 ->  7-ceiling(7/9)  = 6,
...
   1 ->  1-ceiling(1/9)  = 0,
so reaching 0 from 12 requires 10 steps;
  13 -> 13-ceiling(13/9) = 11,
  11 -> 11-ceiling(11/9) = 9,
   9 ->  9-ceiling(9/9)  = 8,
   8 ->  8-ceiling(8/9)  = 7,
   7 ->  7-ceiling(7/9)  = 6,
...
   1 ->  1-ceiling(1/9)  = 0,
so reaching 0 from 13 (or more) requires 11 (or more) steps;
thus, 12 is the largest starting value from which 0 can be reached in 10 steps, so a(10) = 12.

Prog

(Magma) a:=[0]; aCurr:=0; for n in [1..56] do aCurr:=Floor(aCurr*9/8)+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), A279077 (k=7), A279078 (k=8), (this sequence) (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: A033060 A060471 A003045 * A029750 A266480 A246080

Adjacent sequences: A279076 A279077 A279078 * A279080 A279081 A279082

Keyword

nonn

Author

Jon E. Schoenfield, Dec 06 2016

Status

approved