A137284 a(0)=1 and a(n) for n > 0 equals the minimal positive integer such that addition of 2^(-a(n)) to Sum_{k = 0,1,...,n-1} 2^(-a(k)) changes only trailing zeros in its decimal representation.

1, 4, 14, 47, 157, 522, 1735, 5764, 19148, 63609, 211305, 701941, 2331798, 7746066, 25731875, 85479439, 283956550, 943283242, 3133519104, 10409325148, 34579029658, 114869050115, 381586724811, 1267603661786, 4210888217270, 13988267873380, 46468020047392

Offset

0,2

Comments

First and last nonzero decimal digits of 2^(-m) appear respectively at the ceiling(m/log_2(10))-th and m-th positions after the point. Hence a(n+1) equals the minimum solution to ceiling(x/log_2(10)) = a(n) + 1, which is x = ceiling(a(n)*log_2(10)).

Links

Table of n, a(n) for n=0..26.

Formula

a(n+1) = ceiling(a(n)*log_2(10)) = ceiling(a(n)*A020862). - Conjectured by R. J. Mathar, proved by Max Alekseyev

Example

Start from 0;
0 + 2^(-1) = 0.5;
0.5 + 2^(-4) = 0.5625 (first digit "5" is equal to the decimal of previous number);
0.5625 + 2^(-14) = 0.56256103515625 (first digits "5625" are equal to the decimals of previous number);
etc.

Crossrefs

Sequence in context: A082574 A289780 A320404 * A365987 A228178 A000908

Adjacent sequences: A137281 A137282 A137283 * A137285 A137286 A137287

Keyword

nonn,base

Author

Paolo P. Lava and Giorgio Balzarotti, Mar 14 2008

Extensions

Edited and extended by Max Alekseyev, May 13 2009

Status

approved