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

Author

Jon E. Schoenfield, Dec 06 2016

Keywords

• nonn

Comments

Inspired by A278586.

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

Examples

			  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.
		

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, ...}?

Programs

Magma

a:=[0]; aCurr:=0; for n in [1..53] do aCurr:=Floor(aCurr*7/6)+1; a[#a+1]:=aCurr; end for; a;

Formula

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