A253057 Complete list of numbers that take three steps to collapse to a single digit in base 3 (written in base 10).

1781, 3239, 3887, 11177, 14821, 33047, 41065, 43981, 98657, 131461, 393901

Offset

1,1

Comments

From Simon Demers, Oct 20 2017: (Start)

This an exceptionally nice finite sequence based on a surprisingly simple but nontrivial rule: collapse the number expressed in base 3 by inserting plus signs and adding, while minimizing the number of steps (applications).

Butler et al. (2014) proved that any number written in base 2 can be collapsed to a single digit in at most two steps. Any number written in base 3 can be collapsed to a single digit in at most two steps except, surprisingly, for the 11 numbers listed in this sequence. One thing that separates base 3 from larger bases is that there are only 11 base-3 numbers that require three applications!

Let m be the sum of the digits in base-3 expansion. Butler et al. (2014) showed that candidates for this sequence must have m < 82.

(End)

In base 3, the terms are written as 2102222, 11102222, 12022222, 120022222, 202022221, 1200022222, 2002022221, 2020022221, 12000022222, 20200022221, 202000022221. - Andrey Zabolotskiy, Oct 20 2017

Links

Table of n, a(n) for n=1..11.

Steve Butler, R. L. Graham, Richard Stong, Collapsing numbers in bases 2, 3, and beyond, in Gathering for Gardner 10, 2014.

Example

In base 3, one of the possible ways to collapse a(1) in three steps is as follows:
2102222 -> 2102 + 222 = 10101 -> 1 + 01 + 01 = 10 -> 1 + 0 = 1.

Crossrefs

Cf. A253058, A253952, A253953, A293929.

Sequence in context: A167282 A237906 A187524 * A204637 A023088 A251086

Adjacent sequences: A253054 A253055 A253056 * A253058 A253059 A253060

Keyword

nonn,fini,full,nice

Author

N. J. A. Sloane, Jan 13 2015

Status

approved